On the Flexural Analysis of Sandwich and Composite Arches through an Isoparametric Higher-Order Model

Transcription

1 On he Flexural Analysis of Sandwich and Composie Arches hrough an Isoparameric Higher-Order Model Sudhakar R. Marur 1 and Tarun Kan 2 Absrac: A higher-order arch model wih seven degrees of freedom per node is proposed o sudy he deep, shallow, hick, and hin composie and sandwich arches under saic loads. The srain field is modeled hrough cubic axial, cubic ransverse shear, and linear ransverse normal srain componens. As he cross-secional warping is accuraely modeled by his heory, i does no require any shear correcion facor. The sress-srain relaionship is derived from an orhoropic lamina in a hree-dimensional sae of sress. The proposed formulaion is validaed hrough models wih various curvaures, aspec raios, boundary condiions, maerials, and loading condiions. DOI: / ASCE EM CE Daabase subjec headings: Sandwich srucures; Composie srucures; Arches; Saic loads. Inroducion Arches and curved beams made of laminaed composie and sandwich maerials are being employed exensively in various indusries such as auomoive, aerospace, energy, medicine, spors, ec., due o heir high siffness o weigh raio, impac srengh, ec. Sysemaic sudy of composie arches under various saic loading condiions is a key o he developmen of design framework for such srucures. Some of he earlier curved elemen formulaions were based on classical heories, which negleced he ransverse shear deformaion. Lee 1969 formulaed a hree-dimensional arch elemen wih six degrees of freedom per node using Casigliano s heorem. Dawe 1974a,b developed various arch elemen models by using differen orders of polynomial such as cubic, quinic, ec., for angenial and radial displacemens and compared he relaive meris of hese elemens. Prahap sudied he arch elemen wih linearly inerpolaed angenial displacemen and cubically inerpolaed radial displacemen for field consisency o address membrane locking Prahap 1985 and variaional correcness Prahap and Shashirekha Many elemens based on he shear deformable heory of Timoshenko 1921 have been repored in he open lieraure. Saleeb and Chang 1987 developed a linear and quadraic isoparameric elemens based on hybrid-mixed formulaion o sudy arch problems. Mode decomposiion echnique had been used for developing a locking free isoparameric curved beam elemen by Solarski and Chiang Choi and Lim 1995 developed wo- and hree-noded curved beam elemens based on assumed srain fields and Timoshenko beam heory. Krishnan and Suresh 1 CSS Foundaion, 313 A4 Wing, Cauvery Block, NGH Complex, Koramangala, Bangalore , India. srmarur@ iibombay.org 2 Deparmen of Civil Engineering, Indian Insiue of Technology, Powai, Mumbai , India. kan@civil.iib.ac.in Noe. This manuscrip was submied on February 9, 2008; approved on November 9, 2008; published online on March 6, Discussion period open unil December 1, 2009; separae discussions mus be submied for individual papers. This paper is par of he Journal of Engineering Mechanics, Vol. 135, No. 7, July 1, ASCE, ISSN /2009/ /$ formulaed a shear deformable curved beam elemen, wih four degrees of freedom per node. Solarski and Belyschko 1982 sudied he role of reduced inegraion in removing membrane locking in curved beam elemens. Also, hey invesigaed 1983 he shear and membrane locking phenomena in isoparameric finie elemens, which are based on displacemen, hybrid-sress, and mixed formulaions. Babu and Prahap 1986 and Prahap and Babu 1986, 1987 offered key insighs as well as he soluion o he locking problem for curved beams by inroducing field-consisen shear and membrane srain inerpolaions; also hey provided a priori error models for locking errors and sress oscillaions and confirmed he same hrough numerical experimenaion. Lee and Sin 1994 repored a locking free curved beam elemen by choosing curvaures as nodal parameers insead of displacemens, and by incorporaing shear and membrane srains ino he oal poenial energy by he force equilibrium equaion. Balasubramanian and Prahap 1989 developed a field consisen higher-order in cubic polynomial was used for angenial, radial, as well as cross-secional roaional degrees of freedom curved beam elemen based on Timoshenko s heory for he analysis of sepped circular arches. Liewka and Rakowski 1998 employed analyical shape funcions using algebraicrigonomeric funcions o develop an exac siffness marix for sudying curved beams wih consan curvaure. Ganapahi e al developed a cubic B-spline based field consisen elemen for he analysis laminaed curved beams. Raveendranah e al proposed a Timoshenko s heory based field consisen wo-noded curved composie beam elemen, which incorporaes flexural, axial, and shear loadings in he plane of he beam. Masunaga 2003 repored a global higher-order arch heory by aking ransverse shear and normal sress componens ino accoun. By expanding he displacemens hrough power series, he derived he fundamenal equilibrium equaions hrough he principle of virual work, for laminaed circular arches. Malekzadeh 2009 sudied hick laminaed composie circular arches under saic loads using Differenial Quadraure Mehod. Cevik 2007 analyzed a seel fiber reinforced aluminum mealmarix laminaed composie arch under hermal loads o explore he effecs of lay up angle, number of layers, sacking 614 / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

2 sequence, ec., on he residual sresses. Kim and Chaudhuri 2009 sudied he large deflecion behavior of symmerically laminaed hin shallow circular arch under a cenral concenraed load using Rayleigh Riz finie elemen mehod wih C 1 coninuiy for he radial deformaion and C 0 coninuiy for he angenial displacemen. A perusal of earlier works would reveal ha he arch formulaions for saic loading condiions had been made eiher wih classical heory or firs-order shear deformable heory of Timoshenko. I is quie well known ha he classical heory would be adequae only for hin secions. While firs-order heory can handle relaively hick secions, i has serious limiaions such as he need for a shear correcion facor Cowper 1966, inabiliy o model he cross secional warping a key facor for sandwich consrucions wih siff facings and weak cores. Also, i canno model he variaion of ransverse displacemen across he hickness, or in oher words, he ransverse normal srain. Thus, he need for a heory capable of accuraely modeling and analyzing deep composie and sandwich arches is quie eviden. This paper presens a Higher-Order Arch Model HOAM, precisely fulfilling ha need. The HOAM models he cross-secional warping hrough a cubic axial srain; considers he variaion of ransverse displacemen across he hickness hrough a linearly varying ransverse normal srain; i also incorporaes ransverse shear srain, varying cubically across he cross secion. This heory does no require any shear correcion facor and employs sandard isoparameric elemens. Is elasiciy marix had been derived from an orhoropic lamina assumed o be in a hree-dimensional sae of sress, in such a way ha even angle ply laminaions can be sudied using one-dimensional elemens. The HOAM is validaed hrough he saic analyses of shallow o deep and hin o hick laminaed arches, wih various boundary condiions. The role of curvaure, aspec raio, and end condiions on he ransverse deformaions of a sandwich and composie arch are invesigaed hrough parameric sudies and suiable conclusions are drawn. Theoreical Formulaion The Higher-Order Arch Model HOAM, based on Taylor s series expansion Lo e al. 1977, can be expressed, for an arch, as follows: u = u 0 + z x + z 2 u 0 * + z 3 x * w = w 0 + z z + z 2 w 0 * z disance from he neural axis o any poin of ineres along he deph of he arch; u 0 and w 0 axial and ransverse displacemens in x-z plane; x face roaion abou he y-axis, as shown in Fig. 1; and u 0 *, x *, z,w 0 * higher-order erms arising ou of Taylor s series expansion and defined a he neural axis. The oal poenial energy of a sysem can be given as = U W 3 U inernal srain energy; and W is he work done by he exernal forces. The same can be expressed as 1 2 W = b u> p> dx 4b u> = u w, = x z xz, = x z xz, p> = p x p z 4c The field variables can be expressed in erms of nodal degrees of freedom as Fig. 1. Arch geomery wih displacemen componens u> = Z d d 5a d = u 0 w 0 x u* 0 * x z w * 0 5b Z d = z z2 z c z z The srain field for an arch Qau 2004 can be expressed as x = 1 1+z/R u,x + w/r z = w,z 6a 6b xz = w,x + u,z u/r 6c R radius of curvaure. Applying he displacemen field from Eqs. 1 and 2 in he above equaions, one ges x = x0 + z 2 * x0 + z x + z 3 * x 7a U = 2 1 dv 4a z = z0 + z z 7b JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 615

3 xz = + z 2 * + z xz + z 3 * xz 7c and can be expressed in marix form as a = x0 x = Z a a + Z b b z = Z xz = Z s s * x0 = u 0,x + w 0 /R u * 0,x + w * 0 /R 11a b = x * x = x,x + z /R * x,x 11b = z0 z = z 2w 0 * 11c s = * xz * xz = w 0,x + x u 0 /R w * 0,x +3 * x u * 0 /R z,x +2u * 0 x /R * x /R 11d 1 1+z/R z Z b = 1+z/R Z a = z 1+z/R 2 z 1+z/R 3 11e 11f Z = 1 z 11g Z s = 1 z 2 z z 3 11h The srains of Eqs can be rewrien in a combined marix form as = Z a Z = Z Z b 12a Z 0 12b Z s = a b s 12c The sress-srain relaionship of an orhoropic lamina in a hree-dimensional sae of sress can be expressed as Jones 1975 o = Q o 13a o = x y z xy yz xz 13b o = x y z xy yz xz 13c and Q is given by Eqs in Appendix I. By seing y, xy, yz equal o zero in Eq. 13a and deriving he remaining sress componens from he same equaion Vinayak e al. 1996, one ges he sress srain relaionship as = C 14a = x z xz 14b C = C 11 C 12 0 C 21 C C 33 14c and he expressions for various C marix elemens are given by Eqs in Appendix II. The inernal srain energy can be evaluaed using Eqs. 12a 12c and 14a 14c as U = 2 1 dv = 2 1 D dx 15a D = b Z CZ dz =b ZaC11Za Z b C 11 Z a Z C 21 Z a Z a C 11 Z b Z a C 12 Z 0 15b Z b C 11 Z b Z b C 12 Z 0 Z C 21 Z b Z C 22 Z Z s C 33 Z dz s 15c Dab Da 0 D ba D bb D b 0 = Daa 15d D a D b D D ss and he expansions of various D marices are given by Eqs , in Appendix III. The exernal work done of Eq. 4b can be modified wih Eq. 5a as W = d P> dx 16a P> = bz d p> 16b =b p x p z zp x z 2 p x z 3 p x zp z z 2 p z 16c This vecor can now be expressed as P> = p x0 p z0 m x0 p* x0 m* x0 m z0 p * z0 17 Now, he oal poenial energy can be resaed wih Eqs. 15a and 16a as Finie-Elemen Modeling = 2 1 D dx d P> dx 18 The displacemens wihin an elemen can be expressed in erms of is nodal displacemens in isoparameric formulaions as 616 / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

4 d = Na e 19 N shape funcion vecor Appendix IV ; and a e vecor conaining nodal displacemen vecors of an elemen wih n nodes and can be expressed as a e = d 1 d 2... d n 20 Similarly, he srains wih in an elemen can be wrien hrough Eqs. 5b and 12c as = Ba B b B B s a e = B a e 21, for a given node n, he srain displacemen marix can be compued as Assembling he siffness and force vecors of all elemens, given by Eq. 27a, one ges he global saic equilibrium of he srucure, which can be solved by using any sandard solver for saic equilibrium equaions Bahe 1982, afer applying suiable boundary condiions, and he deformaions of he srucure can be esimaed. Numerical Experimens Numerical experimens have been carried ou o sudy he performance of he HOAM. This model is firs validaed by comparing is resuls wih hose available in he lieraure, and subsequenly is performance has been carefully sudied for various maerial and geomeric condiions. Deails such as maerial properies, laminaion scheme and end condiions of every problem solved, are given in Table 1. B a = N,x N/R N,x 0 0 N/R n 22 Validaion Experimens B b = 0 0 N,x 0 0 N/R N/R 0 0 n B = N N n N/R N,x N N/R 3N 0 N,x B s = 0 0 N/R 2N 0 N,x N/R 0 0 n By subsiuing Eqs. 19 and 21 in Eq. 18, one ges = 1 2 a e B D B dxa e a e P C + N P> dx P C vecor of nodal concenraed loads of an elemen. Minimizing he oal poenial energy of Eq. 26 wih respec o nodal degrees of freedom, we ge he saic equilibrium equaion as Ka e = F K = B D B dx F = P C + N P> dx The exernal force vecor of Eq. 27c can be expressed as F = P C + w g N P> J P> = b 0 p z p 2 z z 4 p 27a 27b 27c 28a 28b and p z can eiher be he uniformly disribued ransverse load or he cenral ampliude of he sinusoidal load. Firs, a sandard beam problem is analyzed using HOAM. Resuls of a beam of lengh 10, Young s modulus of , v of 0.25, cross-secion deph of and widh of 1.2, and applied load of 10 boh concenraed and uniformly disribued are given in Table 2, in HOAM can be seen o predic resuls quie close o analyical soluions of Gere and Weaver A clamped-free CF arch wih ip load Fig. 2 and a pinched ring PR Fig. 3 are analyzed wih HOAM, and is resuls are presened in Table 3. One can observe close correlaion wih exac soluions and field consisen finie elemen resuls Balasubramanian and Prahap Nex, a 90 circular ring under pure bending Fig. 4 is sudied and he normalized displacemens wih exac soluions for differen R/ raios are presened in Table 4. Also, a circular arch wih cenral concenraed load Fig. 5 is analyzed wih HOAM. The predicions of HOAM are compared wih exac resuls in Table 5. In boh cases, he HOAM compues resuls quie closer o exac soluions. A quarer ring wih a radial load a he ip Fig. 6 and a pinched ring Fig. 3 solved by Lee and Sin 1994 are modeled wih HOAM for various R/ and S/ raios from hin o hick arch cross secions and he resuls normalized wih analyical soluions are given in Tables 6 and 7. The resuls of HOAM agree very well wih hose of analyical soluions. However, he normalized deformaions vary as he arch becomes hicker, primarily because he analyical soluions considered are based on firsorder shear deformaion heory FOST of Timoshenko 1921, which does no capure he cross-secional warping of deeper cross secions as well as he ransverse normal srains, while HOAM incorporaes boh of hem. Three arch problems Figs. 7 a and b, Fig. 3 analyzed by Ganapahi e al are sudied by HOAM, and he resuls are shown in Table 8. Close correlaion wih he earlier sudies can be observed in his problem as well. Nex, a laminaed arch secor subjeced o sinusoidal load Fig. 8 is sudied for various R/ raios, he close correlaion of HOAM resuls wih he spline based elemen and elasiciy soluion can be observed Table 9. The pinched ring Fig. 3 of Kulikov and Plonikova 2004 is modeled wih HOAM and one can observe he perfec mach beween HOAM and his reference, for various R/ raios, in Table 10. Masunaga s arch secor 2003 subjeced o sinusoidal load JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 617

5 Table 1. Daa for Numerical Experimens Boundary condiions for differen suppors Suppor ype a x=0 a x=s Simply suppored SS u 0 =u * 0 =0 u 0 =u * 0 =0 w 0 = z =w * 0 =0 w 0 = z =w * 0 =0 Roller-simply suppored RS u 0 =u * 0 =0 w 0 = z =w * 0 =0 w 0 = z =w * 0 =0 Clamped-clamped CC u 0 =u * 0 = x = * x =0 u 0 =u * 0 = x = * x =0 w 0 = z =w * 0 =0 w 0 = z =w * 0 =0 Clamped-free CF u 0 =u * 0 = x = * 0 =0 All free w 0 = z =w * 0 =0 Pinched-ring PR u 0 =u * 0 = x = * x =0 u 0 =u * 0 = x = * x =0 No. Maerial daa deails Ref. Daa-1 Balasubramanian and Prahap 1989 Daa-1a Fig. 2 E 1 =E 2 =E 3 = GPa lb/in. 2 G 12 =G 23 =G 13 = GPa lb/in. 2 R=254 mm 10 in. S=254 mm 10 in., = v=0.3, b=25.4 mm 1 in., = mm in. P C = N lb BC: CF Daa-1b Fig. 3 E 1 =E 2 =E 3 = GPa lb/in. 2 G 12 =G 23 =G 13 = GPa lb/in. 2 R= mm in. S of quarer ring = mm in., =90 v=0.3125, b=25.4 mm 1 in., = mm in. P C = N 100 lb, Load on quarer ring= N 50 lb BC: PR Daa-2 Fig. 4 E 1 =E 2 =E 3 = GPa lb/in. 2 G 12 =G 23 =G 13 = GPa lb/in. 2 =90 v=0.25, b=30.48 mm 1.2 in. M 0 = N m 1 lb in. BC: CF Exac values: u= MR2 EI 2 1 w= MR2 EI = MR 2EI Solarski and Chiang 1989 Daa-3 Fig. 5 E 1 =E 2 =E 3 = GPa lb/in. 2 G 12 =G 23 =G 13 = GPa lb/in. 2 =315 v=0.3, b=30.48 mm 1.2 in., =3.175 mm in. R= mm in., S= mm in. P C = N 1 lb BC: CC Solarski and Chiang / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

6 Table 1. Coninued. No. Maerial daa deails Ref. Daa-4 Daa-4a Fig. 6 Daa-4b Fig. 3 Daa-5 Daa-5a Figs. 7 a and b Daa-5b Fig. 3 Daa-6 Fig. 8 Daa-7 Fig. 3 E 1 =E 2 =E 3 = GPa lb/in. 2 G 12 =G 23 =G 13 = GPa lb/in. 2 =90 v=0, b=25.4 mm 1 in. P C = N 50 lb BC: CF Analyical: u= PR 3 /2EI PR/2kGA PR/2EA w= PR 3 /4EI+ PR/4kGA+ PR/4EA = PR 2 /EI P C = N 100 lb, Load on quarer ring= N 50 lb BC: PR Res are he same as Daa-4a Analyical: w 1 = 4 PR 3 /2 EI PR/2kGA+ PR/2EA w 2 = 2 8 PR 3 /4 EI+ PR/4kGA+ PR/4EA E 1 =E 2 =E 3 = KPa 12 lb/in. 2 G 12 =G 23 =G 13 = KPa 4.8 lb/in. 2 R=2540 mm 100 in. S= mm in., =180 v=0.25, b=25.4 mm 1 in. P C = N 1 lb S/ =10,100 BC: CC, CF NDP: w*ei/ P C R 3 S of quarer ring = mm in., =90 P C = N 1 lb, Load on quarer ring= N 0.5 lb BC: PR NDP: w*2ei/ P C R 3 Res are he same as Daa-5a E 1 = KPa 25 lb/in. 2, E 2 =E 3 = KPa 1 lb/in. 2 G 12 = KPa 0.5 lb/in. 2, G 23 =1.379 KPa 0.2 lb/in. 2 G 13 = KPa 0.5 lb/in. 2 R=2540 mm 100 in. S= mm in., =60 v=0.25, b=25.4 mm 1 in. P S = KPa 1 lb/in. 2 BC: RS NDP: w*10e 1 3 / P S R 4 Laminaion: 0/90/0 E 1 =E 2 =E 3 = GPa lb/in. 2 G 12 =G 23 =G 13 = GPa lb/in. 2 R=2540 mm 100 in. v=0.3, b=25.4 mm 1 in. S of quarer ring = mm in., =90 Lee and Sin 1994 Ganapahi e al Ganapahi e al Kulikov and Plonikova 2004 JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 619

7 Table 1. Coninued. No. Maerial daa deails Ref. P C = N 100 lb, load on quarer ring= N 50 lb BC: PR Exac: w=0.8927p C R 3 /E 3 Daa-8 Fig. 8 Maerial daa sandwich Maerial daa composie Daa-9 Fig. 6 Daa-9a Daa-9b Daa-9c Daa-9d E 1 =144.8 GPa, E 2 =E 3 =9.65 GPa G 12 =4.14 GPa, G 23 =3.45 GPa, G 13 =3.45 GPa S=100 m v=0.3, b=1 m P S =1 N/m 2 BC: RS NDP: w*e 1 3 / P S S 4 Laminaion: 0/90, 0/90/0 Face: Graphie/Epoxy E x = GPa lb/in. 2 E y =E z = GPa lb/in. 2 G xy =G yz =G xz = GPa lb/in. 2 kg xy =kg yz =kg xz = GPa lb/in. 2 v f =0.3 Core: Aluminium honeycomb 0.25 in. cell size, in. foil E x =E y =E z =G xy =v c =0 G yz = MPa lb/in. 2 G xz = MPa lb/in. 2 kg yz = MPa lb/in. 2 kg xz = MPa lb/in. 2 c / f =8 E x = GPa lb/in. 2 E y =E z = GPa lb/in. 2 G xy =G yz =G xz = GPa lb/in. 2 kg xy =kg yz =kg xz = GPa lb/in. 2 v=0.25 Maerial daa composie =90 b=25.4 mm 1 in., =25.4 mm 1 in. P C = N 50 lb BC: CF Laminaion: 30/ 30/30 Laminaion: 0/45/ 45/90 Res are he same as Daa-9a Maerial daa sandwich =90 b=25.4 mm 1 in. f op,bo =2.54 mm 0.1 in., c =20.32 mm 0.8 in. P C = N 50 lb BC: CF Laminaion: 0/30/45/60/core/60/45/30/0 Laminaion: 0/90/core/0/90 Res are he same as Daa-9c Masunaga 2003 Chen and Sun 1985 Allen 1969 Reddy / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

8 Table 1. Coninued. No. Maerial daa deails Ref. Daa-10 Fig. 3 Daa-10a Maerial daa composie =90 b=25.4 mm 1 in., =25.4 mm 1 in. P C = N 100 lb, load on quarer ring= N 50 lb BC: PR Laminaion: 30/ 30/30 Daa-10b Laminaion: 0/45/ 45/90 Res are he same as Daa-10a Daa-10c Maerial daa Sandwich =90 b=25.4 mm 1 in. f op,bo =2.54 mm 0.1 in., c =20.32 mm 0.8 in. P C = N 100 lb, load on quarer ring= N 50 lb BC: PR Laminaion: 0/30/45/60/core/60/45/30/0 Daa-10d Laminaion: 0/90/core/0/90 Res are he same as Daa-10c Daa-11 Fig. 9 Daa-11a Maerial daa composie =180 R=2540 mm 100 in., S= mm in. b=25.4 mm 1 in., S/=5 P C = N 50 lb, M 0 = N m 50 lb in. P U = KN/m 50 lb/in., P S = KPa 50 lb/in. 2 Laminaion: 30/ 30/30 Daa-11b Laminaion: 0/45/ 45/90 Res are he same as Daa-11a Daa-11c Maerial daa sandwich =180 R=2540 mm 100 in., S= mm in. b=25.4 mm 1 in., S/=5 f op,bo = mm in., c = mm in. P C = N 50 lb, M 0 = N m 50 lb in. P U = KN/m 50 lb/in., P S = KPa 50 lb/in. 2 Laminaion: 0/30/45/60/core/60/45/30/0 Daa-11d Laminaion: 0/90/core/0/90 Res are he same as Daa-11c Fig. 8 is modeled wih HOAM for symmeric and unsymmeric laminaion schemes as in Table 11. The HOAM quie marginally underpredics for unsymmeric laminaes and over predics for he symmeric laminaes, compared o Masunaga s formulaion. These examples have been carefully chosen in order o capure he variaion in curvaure, aspec raio, and end condiions, on he response of he arch o saic loads. The accuracy and adequacy of HOAM are validaed hrough he good agreemen observed beween he HOAM and he earlier works, which were repored in he lieraure. HOAM Experimens Now, differen arch configuraions such as quarer ring, pinched ring, and a circular arch are sudied using HOAM for differen laminaions, loading, and end condiions. JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 621

9 Table 2. Performance Evaluaion of a Beam wih HOAM Analyical Beam ype Loading Response Gere and Weaver 1965 HOAM HOAM/ analyical Simply suppored Uniformly disribued Deflecion Roaion Poin load a cener Deflecion Roaion Canilever Poin load a free end Deflecion Firs, a clamped-free arch wih a ip load Fig. 6 is sudied for various laminaion schemes and aspec raios, and he resuls are presened in Table 12. In he case of symmeric and unsymmeric composies, HOAM is more flexible compared o FOST. Similarly, u 0 and w 0 predicions of HOAM for sandwich arches, are higher wice or more for deeper secions han hose of Fig. 2. Clamped-free arch segmen wih a radial ip load Fig. 3. Pinched ring and is quarer symmery model FOST; however, he x of HOAM is lower han ha of FOST. The unsymmeric configuraion, for boh composie and sandwich arches, is siffer compared o is symmeric counerpar, for his problem. Nex, a laminaed pinched ring Fig. 3 is analyzed for various aspec raios and he ransverse displacemens a he poin of loading are given in Table 13. The HOAM is flexible for unsymmeric composies and sandwiches compared o FOST. While HOAM is siffer for symmeric composies wih he given angle ply lay up, i becomes flexible as he laminaion ends owards cross-ply configuraion. Also, he order of difference is quie significan in he case of sandwiches HOAM resuls are more han hrice of FOST. Transverse deformaions a he cenral loading poin of a deep circular arch Fig. 9 wih four ypes of load cases and end condiions are analyzed using HOAM and compared wih hose of FOST in Table 14. As one can observe, he HOAM is flexible compared o FOST and paricularly for sandwiches, he order of difference is quie subsanial. In he case of symmeric composies wih simply suppored SS and clamped-clamped CC end condiions, HOAM predics siffer resuls for he given angle ply lay up and becomes flexible as he laminaion ends owards cross-ply configuraion. Similarly, for clamped-free CF sandwich arches wih cenral momen, we find HOAM o be siffer han FOST; however, as he locaion of applicaion of momen moves eiher owards he free end or he simply suppored end, HOAM becomes flexible compared o FOST. On comparing he symmeric and unsymmeric laminaes, one finds he unsymmeric laminaes o be siffer. However, in he case of higher-order composie arches wih roller-simply suppored RS end condiion, symmeric laminaes are siffer. The HOAM resuls of a circular arch under differen ypes of loading, laminaions, and boundary condiions are given in Table 15. The CC end condiion produces siffer arch han SS end condiion. RS end condiion produces perhaps he mos flexible sysem followed by CF condiion. Table 3. Tip and Cenral Deflecions Daa-1 Source Tip deflecion of a clamped-free arch Daa-1a Source Cenral deflecion of a pinched ring Daa-1b Exac Kikuchi and Tanizawa Babu and Prahap Balasubramanian and Prahap Balasubramanian and Prahap HOAM HOAM / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

10 Fig. 4. Quarer ring wih end momen Table 4. Normalized Free end Displacemens of a Canilever Arch under Pure Bending Daa-2 HOAM/exac Solarski and Chiang 1989 R/ u w Fig. 6. Quarer ring wih a radial load a ip Table 6. Normalized Tip Deformaions of a Quarer Ring Daa-4a Aspec raio HOAM/analyical Lee and Sin 1994 R/ S/ u w 100, pi , pi , pi pi pi pi pi pi pi pi pi Table 7. Normalized Tip Deformaions of a Pinched Ring Daa-4b Aspec raio HOAM/analyical Lee and Sin 1994 Fig. 5. Clamped-clamped arch wih cenral load Table 5. Cenral Deflecion of a Circular Arch under Concenraed Force Daa-3 Verical displacemen Source a loading poin Solarski and Belyschko Solarski and Chiang HOAM R/ S/ w 1 w 2 1,000, pi , pi , pi , pi pi pi pi pi pi pi pi pi JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 623

11 Fig. 7. a Circular clamped-clamped arch wih cenral load. b Circular clamped-free arch wih ip load Table 8. Normalized Transverse Deformaions a he Loading Poins Daa-5a, 5b S/ HOAM Ganapahi e al Clamped-clamped circular arch Clamped-free circular arch Pinched ring Table 10. Normalized Transverse Deformaions a he Loading Poin of a Pinched Ring Daa-7 HOAM/analyical Kulikov R/ and Plonikova Table 11. Normalized Transverse Deformaions of a Laminaed Arch Daa-8 S/ S/ R HOAM Masunaga / /90/ Parameric Sudies In order o sudy he influence of curvaure, aspec raio, and end condiions on he deformaions, a circular arch wih a cenral concenraed load P C wih Daa-11c for symmeric sandwich and Daa-11b for unsymmeric composie laminaion, is aken up. The ransverse deformaions a he cener of a sandwich circular arch, he concenraed load is applied for various S/ raios, are shown in Figs The CC end condiion is he siffes, followed by SS and CF, as one would expec. The arch wih CC and SS end condiions demonsraes a similar response paern: i becomes siffer wih he increase in inernal angle up o 180 and hereafer becomes sofer, for he same applied load. In he case of CF end condiion, he srucure becomes siffer wih he increase in inernal angle. Fig. 8. Arch segmen wih sinusoidal load Table 9. Normalized Transverse Cenral Deformaions of a Laminaed Arch Daa-6 R/ HOAM Ganapahi e al Elasiciy Ganapahi e al Fig. 9. Circular arch wih various end and load condiions 624 / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

12 Table 12. Deformaions a he Loading Poin of a Quarer Ring Daa-9 Aspec raio u w R/ S/ HOAM FOST HOAM FOST HOAM FOST Daa 2 pi Sym comp: pi Daa-9a pi pi pi pi Unsym comp: pi Daa-9b pi pi pi pi Sym sandwich: pi Daa-9c pi pi pi pi Unsym sandwich: pi Daa-9d pi pi pi Table 13. Transverse Deformaions a he Loading Poin of a Pinched Ring Daa-10 Aspec raio w R/ S/ HOAM FOST Daa 2 pi Sym comp: pi Daa-10a pi pi pi pi Unsym comp: pi Daa-10b pi pi pi Pi Sym sandwich: pi Daa-10c pi pi pi Pi Unsym sandwich: pi Daa-10d pi pi pi JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 625

13 Table 14. Transverse Deformaions a he Cener of a Circular Arch Daa-11 SS CC CF RS Load ype HOAM FOST HOAM FOST HOAM FOST HOAM FOST Daa Concen Sym comp: Momen Daa-11a udl Sinusoidal Concen Unsym comp: Momen Daa-11b udl Sinusoidal Concen Sym Momen sandwich: udl Daa-11c Sinusoidal Concen Unsym Momen sandwich: udl Daa-11d Sinusoidal Table 15. Deformaions a he Cener of a Circular Arch Daa-11 Sym Comp: Daa-11a Unsym Comp: Daa-11b Load ype and BC u w u w SS-Con SS-Mom SS-udl SS-Sin CC-Con CC-Mom CC-udl CC-Sin CF-Con CF-Mom CF-udl CF-Sin RS-Con RS-Mom RS-udl RS-Sin Sym. Sandwich: Daa-11c Unsym. Sandwich: Daa-11d SS-Con SS-Mom SS-udl SS-Sin CC-Con CC-Mom CC-udl CC-Sin CF-Con CF-Mom CF-udl CF-Sin RS-Con RS-Mom RS-udl RS-Sin / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

14 Fig. 10. Sandwich arch wih concen. load S/=5 Daa-11c Fig. 14. Sandwich arch wih concen. load S/=50 Daa-11c Fig. 11. Sandwich arch wih concen. load S/=10 Daa-11c Fig. 15. Composie arch wih concen. load S/=5 Daa-11b Fig. 12. Sandwich arch wih concen. load S/=15 Daa-11c Fig. 16. Composie arch wih concen. load S/=10 Daa-11b Fig. 13. Sandwich arch wih concen. load S/=25 Daa-11c Fig. 17. Composie arch wih concen. load S/=15 Daa-11b JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 627

15 Fig. 18. Composie arch wih concen. load S/=25 Daa-11b Fig. 20. Axis sysem 1,2,3: Lamina axes; x,y,z: laminae axes Fig. 19. Composie arch wih concen. load S/=50 Daa-11b The responses of an unsymmeric composie arch are presened in Figs The arch wih SS end condiion urns siff quie seeply wih he increase in he subended angle. This inernal angle a which he ransiion happens also changes wih aspec raio. The CC arch, which is he siffes, also follows a similar paern. The CF arch becomes siffer wih he increase in inernal angle. Conclusions A higher-order arch model HOAM wih ransverse shear and normal srain componens is formulaed for sudying he response of laminaed arches under various saic loading condiions. The HOAM can sudy shallow o deep and hin o hick arch geomeries quie effecively. Through he consiuive relaionship, adaped from he hree-dimensional sress-srain relaionship of an orhoropic lamina, even angle-ply laminaes can be analyzed using one-dimensional elemens. The HOAM is validaed hrough he resuls of earlier invesigaions. Subsequenly, he response of HOAM for various maerials, loading, boundary, and geomeric condiions are presened and also compared wih hose of firs-order shear deformaion heory FOST of Timoshenko. Parameric sudies on symmeric sandwich and unsymmeric composie arches have been carried ou o sudy he role of curvaure, aspec raio, and end condiions on he ransverse deformaions. Appendix I. Coefficiens of Q-Marix The sress-srain relaionship a a poin of an orhoropic lamina in a hree dimensional sae of sress/ srain can be expressed Jones 1975, along he lamina axes Fig. 20 as = D 29 = = E E E E E E D = 1 E E E G G G 13 = = R s 34 The relaion beween engineering and ensor srain vecors, along lamina and laminae axes, can be given as = R s / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

16 R = If he angle beween lamina and laminae axes can be defined as, hen he lamina o laminae axis ransformaion is given by = c2 T s2 0 2sc 0 0 s 2 c 2 0 2sc sc sc 0 c 2 s c s s c Q 24 = D 11 D 12 2D 44 s 3 c + D 12 D 22 +2D 44 sc 3 Q 33 = D 33 Q 34 = D 13 D 23 sc Q 44 = D 11 2D 12 + D 22 2D 44 s 2 c 2 + D 44 c 4 + s 4 Q 55 = D 55 c 2 + D 66 s 2 Q 56 = D 66 D 55 sc Q 66 = D 55 s 2 + D 66 c 2 Appendix II. Coefficiens of C-Marix c = cos s = sin 38 and he sress and srain along he lamina and laminae axes can be equaed as = T 39 s = T s 40 By making use of Eqs , one can ge he laminae sresssrain relaionship as = Q 22 Q 44 Q C 11 = Q 11 + Q 12 Q 14Q 24 Q 12 Q 44 + Q 14 Q 12Q 24 Q 14 Q 22 C 12 = Q 13 + Q 12 Q 24Q 34 Q 23 Q 44 + Q 14 Q 23Q 24 Q 22 Q = Q Q = T 1 D T C 21 = Q 13 + Q 23 Q 14Q 24 Q 12 Q 44 + Q 34 Q 12Q 24 Q 14 Q T 1 = RTR 1 43 Q12 Q13 Q Q 21 Q 22 Q 23 Q Q 31 Q 32 Q 33 Q Q = Q11 44 Q 41 Q 42 Q 43 Q Q 55 Q Q 65 Q 66 C 22 = Q 33 + Q 23 Q 24Q 34 Q 23 Q 44 + Q 34 Q 23Q 24 Q 22 Q 34 C 33 = Q 66 Q 2 56 Q Q 11 = D 11 c 4 +2 D 12 +2D 44 s 2 c 2 + D 22 s 4 Q 12 = D 12 s 4 + c 4 + D 11 + D 22 4D 44 s 2 c Appendix III. Coefficiens of D-Marix Using a binomial series, he following erms can be expanded as Q 13 = D 31 c 2 + D 32 s z/R =1 z R + z2 R 2 z3 R 3 64 Q 14 = D 11 D 12 2D 44 sc 3 + D 12 D 22 +2D 44 s 3 c Q 22 = D 11 s 4 +2 D 12 +2D 44 s 2 c 2 + D 22 c z/R 2 =1 2z R + 3z2 R 2 4z3 R 3 65 Q 23 = D 13 s 2 + D 23 c 2 50 which are used in he evaluaion of various D marices JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 629

17 2 NL 1 D aa = b Z a C 11 Z R a dz = b C 11 H H R 2H 3 4 R 3H 4 H 3 2 R H R 2H 5 4 R 3H 8 6 l=1 H 3 2 R H R 2H 5 4 R 3H 6 H 5 2 R H R 2H R 3H 2 NL 2 D ab = b Z a C 11 Z R b dz = b C 11 H H R 2H 4 4 R 3H 5 H 4 2 R H R 2H 6 4 R 3H 9 7 l=1 H 4 2 R H R 2H 6 4 R 3H 7 H 6 2 R H R 2H R 3H D ba = D ab 68 2 NL 3 D bb = b Z b C 11 Z R b dz = b C 11 H H R 2H 5 4 R 3H 6 H 5 2 R H R 2H 7 4 R 3H 8 l=1 H 5 2 R H R 2H 7 4 R 3H 8 H 7 2 R H R 2H R 3H 2 NL 1 D a = b Z a C 12 Z R dz = b C 12 H H R 2H 3 4 R 3H 4 H 2 2 R H R 2H 4 4 R 3H 7 5 l=1 H 3 2 R H R 2H 5 4 R 3H 6 H 4 2 R H R 2H R 3H 2 NL 2 D b = b Z b C 12 Z R dz = b C 12 H H R 2H 4 4 R 3H 5 H 3 2 R H R 2H 5 4 R 3H 8 6 l=1 H 4 2 R H R 2H 6 4 R 3H 7 H 5 2 R H R 2H R 3H 2 NL 1 D a = b Z C 21 Z R a dz = b C 21 H H R 2H 3 4 R 3H 4 H 3 2 R H R 2H 5 4 R 3H 7 6 l=1 H 2 2 R H R 2H 4 4 R 3H 5 H 4 2 R H R 2H R 3H 2 NL 2 D b = b Z C 21 Z R b dz = b C 21 H H R 2H 4 4 R 3H 5 H 4 2 R H R 2H 6 4 R 3H 8 7 l=1 H 3 2 R H R 2H 5 4 R 3H 6 H 5 2 R H R 2H R 3H NL D = b Z C 22 Z dz = b C l=1 22 H 1 H 2 H 2 H 3 74 NL D ss = b Z s C 33 Z s dz = b l=1 In Eqs , for a given layer l 33 H1 H3 H2 H4 C H 5 H 4 H 6 7 H 3 H 5 Sym H 75 H p = 1 p h l p h p l 1 76 NL oal number of layers of a cross secion; p consan varying from 1 o 10; h l disance from he neural axis o he op of a layer, l, and h l 1 disance from he neural axis o he op of layer l 1 or boom of layer l. Appendix IV. Shape Funcions The shape funcions of isoparameric elemens used in his sudy are presened as follows. Linear Elemen N 1 = 1.0 /2 N 2 = / / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009

18 Quadraic Elemen N 1 = + 2 /2 N 2 = N 3 = + 2 /2 Cubic Elemen N 1 = /16 Noaion N 2 = /16 N 3 = /16 N 4 = /16 The following symbols are used in his paper: BC boundary condiion; FOST firs-order shear deformaion heory of Timoshenko 1921 ; HOAM higher-order arch model; k shear correcion facor=5/6; M 0 momen applied on he srucure; NDP nondimensionalizing parameer; and P C concenraed load; magniude of sinusoidal load P S sin x/s ; P S P U uniformly applied load; R radius of curvaure; S arc lengh of he arch; S/ aspec raio; hickness of cross secion; and =S/R subended angle of an arch. References Allen, H. G Analysis and design of srucural sandwich panels, Pergamon Press, London. Babu, C. R., and Prahap, G A linear hick curved beam elemen. In. J. Numer. Mehods Eng., 23 7, Balasubramanian, T. S., and Prahap, G A field consisen higher-order curved beam elemen for saic and dynamic analysis of sepped arches. Compu. Sruc., 33 1, Bahe, K. J Finie elemen procedures in engineering analysis, Prenice-Hall, N.J. Cevik, M An elasic-plasic hermal sress analysis of cross-ply and angle-ply aluminum meal-marix composie arches. J. Reinf. Plas. Compos., 26 2, Chen, J. K., and Sun, C. T Nonlinear ransien responses of iniially sressed composie plaes. Compu. Sruc., 21 3, Choi, J. K., and Lim, J. K General curved beam elemens based on he assumed srain fields. Compu. Sruc., 55 3, Cowper, G. R The shear coefficien in Timoshenko beam heory. J. Appl. Mech., 33 2, Dawe, D. J. 1974a. Curved finie elemens for he analysis of shallow and deep arches. Compu. Sruc., 4 3, Dawe, D. J. 1974b. Numerical sudies using circular arch finie elemens. Compu. Sruc., 4 4, Ganapahi, M., Pael, B. P., Saravanan, J., and Touraier, M Shear flexible curved spline beam elemen for saic analysis. Finie Elem. Anal. Design, 32 3, Gere, J. M., and Weaver, W Analysis of Framed Srucures, D. Van Nosrand Company, New York, 450. Jones, R. M Mechanics of composie maerials, McGraw-Hill Kogakusha, Tokyo. Kikuchi, F., and Tanizawa, K Accuracy and locking free propery of beam elemen for arch problems. Compu. Sruc., , Kim, D., and Chaudhuri, R. A Pos buckling behavior of symmerically laminaed hin shallow circular arches. Compos. Sruc., 87 1, Krishnan, A., and Suresh, Y. J A simple cubic linear elemen for saic and free vibraion analyses of curved beams. Compu. Sruc., 68 5, Kulikov, G. M., and Plonikova, S. V Non-convenional nonlinear wo-node hybrid sress-srain curved beam elemens. Finie Elem. Anal. Design, 40 11, Lee, H. P Generalized siffness marix of a curved-beam elemen. AIAA J., 7 10, Lee, P. G., and Sin, H. C Locking-free curved beam elemen based on curvaure. In. J. Numer. Mehods Eng., 37 6, Liewka, P., and Rakowski, J The exac hick arch finie elemen. Compu. Sruc., 68 4, Lo, K. H., Chrisensen, R. M., and Wu, E. M A higher-order heory of plae deformaion. Par 1: Homogenous plaes. J. Appl. Mech., 44 4, Malekzadeh, P A wo-dimensional layerwise-differenial quadraure saic analysis of hick laminaed composie circular arches. Appl. Mah. Model., 33 4, Masunaga, H Inerlaminar sress analysis of laminaed composie and sandwich circular arches subjeced o hermal/ mechanical loading. Compos. Sruc., 60 3, Prahap, G The curved beam/deep arch/finie ring elemen revisied. In. J. Numer. Mehods Eng., 21 3, Prahap, G., and Babu, C. R An isoparameric quadraic hick curved beam elemen. In. J. Numer. Mehods Eng., 23 9, Prahap, G., and Babu, C. R Field-consisency and violen sress oscillaions in he finie elemen mehod. In. J. Numer. Mehods Eng., 24 10, Prahap, G., and Shashirekha, B. R Variaionally correc assumed srain field for he simple curved beam elemen. Compu. Sruc., 47 6, Qau, M. S Vibraion of laminaed shells and plaes, Elsevier, Oxford. Raveendranah, P., Singh, G., and Pradhan, B Applicaion of coupled polynomial displacemen fields o laminaed beam elemens. Compu. Sruc., 78 5, Reddy, J. N On he soluions o forced moions of recangular composie plaes. J. Appl. Mech., 49 3, Saleeb, A. F., and Chang, T. Y On he hybrid-mixed formulaion of C 0 curved beam elemens. Compu. Mehods Appl. Mech. Eng., 60 1, Solarski, H., and Belyschko, T Membrane locking and reduced inegraion for curved elemens. J. Appl. Mech., 49 1, Solarski, H., and Belyschko, T Shear and membrane locking in curved C 0 elemens. Compu. Mehods Appl. Mech. Eng., 41 3, Solarski, H. K., and Chiang, M. Y. M The mode decomposiion C 0 formulaion of curved wo-dimensional srucural elemens. In. J. Numer. Mehods Eng., 28 1, Timoshenko, S. P On he correcion for shear in differenial equaion for ransverse vibraions of prismaic bars. Philos. Mag., 41 6, Vinayak, R. U., Prahap, G., and Naganarayana, B. P Beam elemens based on a higher-order heory. I: Formulaion and analysis of performance. Compu. Sruc., 58 4, JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2009 / 631