Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV BUCKLING OF CYLINDRICAL INFLATABLE COMPOSITE BEAMS USING ISOGEOMETRIC ANALYSIS MẤT ỔN ĐỊNH CỦA DẦM HƠI HÌNH TRỤ SỬ DỤNG PHƯƠNG PHÁP ĐẲNG HÌNH HỌC Thu Phan-Thi-Dang1a, Tuan Le-Manh1b, Giang Le-Hieu2c, Truong Nguyen-Thanh3,4d Faculty of Mechanical Engineering, Ho Chi Minh City Vocational College HCMC University of Technology and Education Industrial Maintenance Training Center (IMTC), Ho Chi Minh City University of Technology (HCMUT) a dangthu0511@yahoo.com; bmanhtuanle@gmail.com c gianglh@hcmute.edu.vn; dthtruong@hcmut.edu.vn ABSTRACT In this work, stability of Timoshenko cylindrical inflatable composite beams is investigated based on isogeometric analysis (IGA) Linear term of Green-Lagrange strain tensor is employed in considering linear eigen buckling analysis of the beam Finite element mesh in C1 continuity is obtained by using quadratic NURBS (Non-Uniform Rational BSpline) elements Numerical model of a cantilever inflatable beam subjected to different inflation pressures is constructed to derive corresponding critical loads Convergence test illustrates the advantages of the approach over the previous theories based on standard finite element methods Keywords: inflatable composite beam, linear eigen buckling, isogeometric analysis TÓM TẮT Trong báo này, ổn định dầm composite Timoshenko bơm khí hình trụ nghiên cứu dựa phương pháp giải tích đẳng hình học Thành phần tuyến tính tensor biến dạng Green-Lagrange sử dụng để xem xét phân tích toán ổn định trị riêng tuyến tính Lưới phần tử hữu hạn đạt liên tục C1 sử dụng phần tử NURBS bậc hai Mô hình số dầm đầu ngàm đầu tự do, áp lực bơm khí khác xây dựng để tìm tải tới hạn tương ứng Phân tích kiểm tra hội tụ thể ưu điểm phương pháp so với phương pháp phần tử hữu hạn truyền thống Từ khóa: dầm bơm khí, ổn định, giải tích đẳng hình học INTRODUCTION In the recent decades, composite materials are extensively applied in industry varying from large scale structures Structural applications with inflatable beams or arches with modern textile materials have been growing and requiring a great effort on theoretical development The theories of the orthotropic inflatable beams were presented [1-10] Le van and Wielgosz [2, 3] have proven that for highly inflated isotropic beams, a linear model is sufficient to get the static and dynamic responses of these structures Finite element analyses of inflated fabric structures present a challenge in that both material and geometric on linearities arise due to the nonlinear load/deflection behavior of the fabric (at low loads), pressure stiffening of the inflated fabric, fabric-tofabric contact, and fabric wrinkling on the structure surface In addition to checking fabric loads, the finite element model is used to predict the fundamental mode of the inflated fabric beam In this paper, numerical model of 821 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV cylindrical inflatable beam is developed in the isogeometric analysis framework Quadratic NURBS-based Timoshenko beam elements are employed in convergence test The objective is to extend the linear model This paper uses all assumptions and development made in the previous paper presented by Thanh-Truong Nguyen et al [4] Furthermore, the inflation pressure prestressing tensor is still assumed spheric and isotropic NURBS-BASED ISOGEOMETRIC ANALYSIS 2.1 Basis function A knot vector is firstly defined as a set of non-decreasing real numbers Ξ = {ξ1 , ξ , , ξ n + p +1} Uniform open knot vector usually used in mechanics has the first and the last knot repeating p + times and the others in uniform space, where p ≥ is the order of B-Spline polynomials, also called B-Spline basis functions, are constructed as follows, 1 , if ξi ≤ ξ ≤ ξi +1 N i ,0 (ξ ) =  , otherwise 0 ξ −ξ ξ − ξi N i , p −1 (ξ ) + i + p +1 N i +1, p −1 (ξ ) ξi + p − ξi ξ i + p +1 − ξ i +1 = N i , p (ξ ) (1a) (1b) where i = 1,2, n B-Spline curve is a tensor product of a set of n control points, { Bi } ∈ R d , and B-Spline basis functions, n C (ξ ) = ∑ N i , p (ξ ) Bi ( x, y, z ) (2) i =1 NURBS curve is obtained by projecting a nonrational B-Spline curve C w (ξ ) from a homogeneous coordinate in R d +1 space onto physical R d space, has a final form like Eq (2) with the basis is Ri , p (ξ ) = N i , p (ξ ) wi n ∑ N (ξ ) w i =1 i, p (3) i where wi ≥ is the weight of control point Biw in homogeneous coordinate 2.2 Isogeometric analysis model Isogeometric analysis proposed by Hughes et al [11] in 2005 using NURBS basis to construct exact geometry and finite element interpolating functions has received numerous attentions More accurate solutions, compared with standard finite element, are usually obtained due to the higher-order continuity in the NURBS mesh In finite element subdomain, dependent displacements, and initial geometric information could be described as follows, nCe s e = ∑ Rce sce (4) c =1 where sce is displacements of control points, or control parameters in homogeneous space nCe is number of control point per element 822 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV THEORETICAL FORMULATION 3.1 Kinematics In Timoshenko beam theory, the displacement field is assumed as, u x ( x= , z) u ( x) + zφ ( x) u z ( x, z) = w( x) (5) where ( u x , u z ) and ( u , w ) are the displacements of a point in body and displacements of a point on midplane along ( x, z ) coordinates, respectively, φ is the rotations of transverse normal of the mid-plane about the y axis, respectively Linear terms of Green-Lagrange strain tensor are employed in considering linear eigen buckling analysis of the beam 3.2 Finite element formulations 3.2.1 Linear eigen buckling In case of linear buckling analysis, the beam is subjected to the inflation pressure prestressing S o tensor The first step is to load the inflated beam by an arbitrary reference level of external load, {Fref } and to perform a standard linear analysis to determine the finite element stresses in the beam It is desirable to also have a general formula for finite element stress stiffness matrix [k σ ] and finite element conventional elastic stiffness matrix [k ] The strain energy of a finite inflatable beam is: { } o T S E + ET ⋅ C ⋅ E dVo = U m + U b ( ) ∫ V o Ue = (6) where U m is the change in membrane energy and U b is the strain energy in bending The strain energy component U m is associated with the stress stiffness matrix [k σ ] of the beam and U b relates to the conventional elastic stiffness [k ] of the beam, as Um = T d  [k σ ][d ] 2  (7) Ub = T d  [k ][d ] 2  (8) By applying the discretization procedure, Ue = ( ) T {d} [k ] + λ k ref  {d} (9) where λ is the proportionality coefficient such as F = λ Fref , with F is the axial load The coefficients in matrices [k ] and k ref  are constant and only are dependent on the geometry, material properties and the inflation pressure prestressing conditions acting on the beam The potential energy for the whole structure can be expressed as = U ( ) T {D} [K ] + λ K ref  {D} (10) Let buckling displacements {δ D} take place relative to displacements {D} of the reference configuration The structural equilibrium equations can be obtained by applying the principle of minimum potential energy This gives an eigenvalue problem in the form: 823 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV ([K ] + λ K  ){δ D} = i ref (11) Eq (11) is an eigenvalue problem where λi is the eigenvalue of first buckling mode The smallest root λcr defines the smallest level of external load for which there is bifurcation, namely {F}cr = λcr {F}ref As the beam is loaded by an arbitrary reference level of external load (12) {F}ref , the of {δ D} is eigenvector {δ D} associated with λcr is the buckling mode The magnitude indeterminate in a linear buckling problem, so that it defines a shape but not an amplitude NUMERICAL MODELS OF TIMOSHENKO INFLATABLE COMPOSITE BEAM Figure 1: Model of a cantilever flatable beam subjected to axial compresion load Fig shows an inflatable cylindrical composite beam l0 , R0 , t0 , A0 and I represent respectively the length, the fabric thickness, the external radius, the cross-section and the moment of inertia around the principal axes of inertia Y and Z of the beam in the reference configuration which is the inflated configuration A0 and I are given by, A0 = 2π R0t0 I0 = A0 R02 where the reference dimensions l0 , R0 and t0 depend on the inflation pressure and the mechanical properties of the fabric pR l lo = lφ + φ φ (1 − 2ν lt ) Et tφ Ro = Rφ + to= tφ − pRφ2 Et tφ pRφ Et ( −ν lt ) ν lt in which lφ , Rφ and tφ are respectively the length, the fabric thickness, and the external radius of the beam in the natural state The internal pressure p is assumed to remain constant, which simplifies the analysis and is consistent with the experimental observations and the prior studies on inflated fabric beams and arches The initial pressurization takes place prior to the application of concentrated and distributed external loads, and is not included in the structural analysis per se The slenderness ratio is λ = L / ρ where L = µ l0 is the beam length and ρ = I / A0 is the beam radius of gyration The coefficient µ takes different values according to the boundary conditions of the beam 824 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV The beam is under- going axial loading Two Fichter’s simplifying assumptions are applied in the following: • The cross-section of the inflated beam under consideration is assumed to be circular and maintains its shape after deformation, so that there are no distortion and local buckling; • The rotations around the principal inertia axes of the beam are small and the rotation around the beam axis is negligible Due to the first assumption, the model considers that no wrinkling occurs so that the ovalization problem is not addressed in this paper Geometric properties: tφ 125 ×10−6 m lφ = 0.65m , Rφ = 0.08m and = Material properties: = E 2.5 ×108 Pa and µ = 0.3 Clamped boundary conditions: u = w = φ = at x = −l0 / Inflation pressures applied to the beam: p1 = ×104 Pa, p2 = 10 ×104 Pa p3 = 15 ×104 Pa, p4 = 20 ×104 Pa Figure 2: Convergence test for buckling load of the cantilever inflatable beam with different inflation pressures Buckling load of the cantilever inflatable beam with different inflation pressures based on isogemetric analysis is plotted in the Fig The obtained results are in excellent agreement with ones derived using standard finite element methods given by Thanh-Truong Nguyen [12] Futhermore, it can be observed a fast convergence in isogeometric analysis due to the high continuity in finite element mesh Addtionally, isogemetric analysis requires less total degrees of freedom (DOFs) than standard FEM and hence saving the computational effort that is significant in nonlinear analysis of the inflatable composite beams 825 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV CONCLUSIONS The linear eigen buckling of the inflatable composite beams is successfully obtained in the framework of NURBS-based isogeometric analysis Timoshenko beam theory and linear term of Green strain tensor are employed for deriving analytical equations Finite element mesh is constructed in C1 continuity by using quadratic NURBS element These reliable solutions verify the accuracy of the proposed method The fast convergence and using less DOFs also show the robustness of the isogeometric analysis inflatable composite beam models that promissing in further analysis of geometric and material nonlinearity REFERENCES [1] Davids, W.G., Zhang, H., 2008 Beam finite element for nonlinear analysis of pressurized fabric beam-columns Engineering Structures 30, 1969–1980 [2] Le Van, A., Wielgosz, C., 2005 Bending and buckling of inflatable beams: some new theoretical results Thin-Walled Structures 43, 1166–1187 [3] Le Van, A., Wielgosz, C., 2007 Finite element formulation for inflatable beams ThinWalled Structures 45, 221–236 [4] Thanh-Truong Nguyen, Van-Quang Huynh, Dinh-Huan Phan, Analytical buckling analysis of an inflatable beam made of orthotropic technical textiles Hội nghị Cơ học toàn quốc lần thứ IX Hà Nội, 8-9/12/2012 [5] Plaut, R.H., Goh, J.K.S., Kigudde, M., Hammerand, D.C., 2000 Shell analysis of an inflatable arch subjected to snow and wind loading International Journal of Solids and Structures 37, 4275–4288 [6] Molloy, S.J., 1998 Finite element analysis of a pair of leaning pressurized arch-shells under snow and wind loads Master’s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA [7] Molloy, S.J., Plaut, R.H., Kim, J.Y., 1999 Behavior of pair of leaning arch-shells under snow and wind loads Journal of Engineering Mechanics 125, 663–667 [8] Veldman, S.L., Bergsma, O.K., Beukers, A., 2005 Bending of anisotropic inflated cylindrical beams Thin-Walled Structures 43, 461–475 [9] Wielgosz, C., Thomas, J.C., 2002 Deflection of inflatable fabric panels at high pressure Thin-Walled Structures 40, 523–536 [10] Wielgosz, C., Thomas, J.C., 2003 An inflatable fabric beam finite element Communications in Numerical Methods in Engineering 19, 307–312 [11] T J R Hughes, J A Cottrell, Y Bazilevs Isogeometric Analysis: CAD, Finite elements, NURBS, Exact Geometry and Mesh Refinement Computer methods in applied mechanics and engineering, 2005, Vol 194(39), pp 4135-4195 [12] Thanh-Truong Nguyen, PhD Dessertation “Numerical modeling and buckling analysis of inflatable structures”, 2012 AUTHOR’S INFORMATION Thu Phan-Thi-Dang: Ho Chi Minh City Vocational College, dangthu0511@yahoo.com, (+84)0903373645 Tuan Le-Manh: Ho Chi Minh City Vocational College, manhtuanle@gmail.com, (+84)0903035206 Giang Le-Hieu: HCMC University of Technology and Education, gianglh@hcmute.edu.vn, (+84)0938308141 Truong Nguyen-Thanh: Industrial Maintenance Training Center (IMTC), Ho Chi Minh City University of Technology (HCMUT), thtruong@hcmut.edu.vn, (+84)0969356839 826