BUCKLING OF CYLINDRICAL INFLATABLE COMPOSITE BEAMS USING ISOGEOMETRIC ANALYSIS mất ổn ĐỊNH của dầm hơi HÌNH TRỤ sử DỤNG PHƯƠNG PHÁP ĐẲNG HÌNH học

BUCKLING OF CYLINDRICAL INFLATABLE COMPOSITE BEAMS USING ISOGEOMETRIC ANALYSIS M ẤT ỔN ĐỊNH CỦA DẦM HƠI HÌNH TRỤ SỬ DỤNG PHƯƠNG PHÁP ĐẲNG HÌNH HỌC 1Faculty of Mechanical Engineering,

BUCKLING OF CYLINDRICAL INFLATABLE COMPOSITE BEAMS USING

ISOGEOMETRIC ANALYSIS

M ẤT ỔN ĐỊNH CỦA DẦM HƠI HÌNH TRỤ SỬ DỤNG PHƯƠNG PHÁP

ĐẲNG HÌNH HỌC

1Faculty of Mechanical Engineering, Ho Chi Minh City Vocational College

2HCMC University of Technology and Education

3Industrial Maintenance Training Center (IMTC),

4Ho Chi Minh City University of Technology (HCMUT)

a dangthu0511@yahoo.com; b manhtuanle@gmail.com

c gianglh@hcmute.edu.vn; d thtruong@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 B-Spline) 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

Trong bài báo này, ổn định của dầm composite Timoshenko bơm khí hình trụ được nghiên c ứu dựa trên phương pháp giải tích đẳng hình học Thành phần tuyến tính của tensor

bi ến dạng Green-Lagrange được sử dụng để xem xét phân tích bài toán mất ổn định trị riêng tuyến tính Lưới phần tử hữu hạn đạt liên tục C 1 do sử dụng phần tử NURBS bậc hai Mô hình

s ố của dầm hơi một đầu ngàm một đầu tự do, dưới áp lực bơm khí khác nhau được xây dựng

để tìm tải tới hạn tương ứng Phân tích kiểm tra hội tụ thể hiện những ưu điểm của phương pháp so v ới các phương pháp phần tử hữu hạn truyền thống

1 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

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

2 NURBS-BASED ISOGEOMETRIC ANALYSIS

2.1 Basis function

A knot vector is firstly defined as a set of non-decreasing real numbers {ξ ξ1, 2, ,ξ + +1}

=

Ξ n p Uniform open knot vector usually used in mechanics has the first and the last knot repeating p+1 times and the others in uniform space, where p≥0 is the order

of B-Spline polynomials, also called B-Spline basis functions, are constructed as follows,

,0

1 ,

0 ,

ξ ξ ξ

= 



i

if N

1 1

ξ ξ

+ +

−

−

i p i

where i=1, 2, n

B-Spline curve is a tensor product of a set of n control points, { } d

i

B ∈R , and B-Spline basis functions,

1

, ,

n

i

C ξ N ξ B x y z

=

NURBS curve is obtained by projecting a nonrational B-Spline curve w( )

C ξ from a homogeneous coordinate in d+ 1

R space onto physical d

R space, has a final form like Eq (2)

with the basis is

( )

, ,

, 1

i

R

ξ ξ

ξ

=

=

where w i ≥ is the weight of control point 0 w

i

B 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,

1

=

=∑nCe

c c c

R

where e

c

s is displacements of control points, or control parameters in homogeneous space nCe is number of control point per element

3 THEORETICAL FORMULATION

3.1 Kinematics

In Timoshenko beam theory, the displacement field is assumed as,

( , z) ( ) ( ) ( , z) ( )

x z

u x u x z x

u x w x

φ

where (u u x, 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 tensor The first step is to load the inflated beam by an arbitrary reference o

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 [ ]kσ and finite element conventional elastic stiffness matrix [ ]k The

strain energy of a finite inflatable beam is:

( )

1

2 o

T

U = ∫ S E E+ ⋅ ⋅C E dV =U +U (6) where U is the change in membrane energy and m U b is the strain energy in bending The strain energy component U is associated with the stress stiffness matrix m [ ]kσ of the beam and U relates to the conventional elastic stiffness b [ ]k of the beam, as

[ ][ ]

1 2

T m

[ ][ ]

1 2

T b

By applying the discretization procedure,

1 2

T

where λ is the proportionality coefficient such as F =λF ref , with F is the axial load The coefficients in matrices [ ]k and kref 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

1 2

T

ref

Let buckling displacements { }δD 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:

[ ]

( K +λiKref) { }δD =0 (11)

Eq (11) is an eigenvalue problem where λ is the eigenvalue of first buckling mode The i smallest root λ defines the smallest level of external load for which there is bifurcation, namely cr

{ }F cr =λcr{ }F ref (12)

As the beam is loaded by an arbitrary reference level of external load { }F ref , the eigenvector { }δD associated with λ is the buckling mode The magnitude of cr { }δD is

indeterminate in a linear buckling problem, so that it defines a shape but not an amplitude

4 NUMERICAL MODELS OF TIMOSHENKO INFLATABLE COMPOSITE BEAM

Figure 1: Model of a cantilever flatable beam subjected to axial compresion load

Fig 1 shows an inflatable cylindrical composite beam l , 0 R , 0 t , 0 A and 0 I represent 0

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 A and 0 I are given by, 0

0 2 0 0

A = πR t

2

0 0 0

2

A R

I =

where the reference dimensions l , 0 R and 0 t depend on the inflation pressure and the 0

mechanical properties of the fabric

2

1 2 2

2 2 3 2

= −

t

t

t

pR l

l l

E t pR

R R

E t pR

t t

E

φ φ φ

φ φ φ

φ φ φ

ν ν ν

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=µl0 is the beam length and ρ = I0/A0 is the beam radius of gyration The coefficient µ takes different values according to the boundary conditions of the beam

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:

0.65

lφ = m, Rφ =0.08m and 6

125 10

tφ = × − m

Material properties:

8 2.5 10

E= × Pa and µ=0.3 Clamped boundary conditions:

0

0 at x / 2

u= = =w φ = −l

Inflation pressures applied to the beam:

5 10 , 10 10

15 10 , 20 10

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 2 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

5 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 Thin-Walled 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

1 Thu Phan-Thi-Dang: Ho Chi Minh City Vocational College, dangthu0511@yahoo.com,

(+84)0903373645

2 Tuan Le-Manh: Ho Chi Minh City Vocational College, manhtuanle@gmail.com,

(+84)0903035206

3 Giang Le-Hieu: HCMC University of Technology and Education, gianglh@hcmute.edu.vn, (+84)0938308141

4 Truong Nguyen-Thanh: Industrial Maintenance Training Center (IMTC), Ho Chi Minh

City University of Technology (HCMUT), thtruong@hcmut.edu.vn, (+84)0969356839