Luận án Phân tích tính ổn định kết cấu dầm bơm hơi vật liệu composite

uadrature for NURBS elements has been investigated and proven as reliable 
in the literature Chawla [1] and Wolfgang [25] as well as in the benchmark examples 
in this thesis (for the examples presented in this thesis, Gauss integration has been 
used). An efficient quadrature for NURBS-based isogeometric analysis that makes 
use of the higher continuities between elements, and therefore is more efficient than 
Gauss quadrature, is developed by Hughes [40]. 
Equivalent to finite elements, a NURBS element is defined by a set of nodes 
and corresponding basis functions. The nodes are the NURBS control points which 
carry the degrees of freedom for the analysis and boundary conditions are applied to 
them. Since the element formulation in this thesis is displacement-based, the degrees 
of freedom are the displacements of the control points. For two-dimensional 
structures this means that every control point has three degrees of freedom, namely 
the displacements in x- and y- direction. It is important to note that with this definition 
3.1 Overview and basics of Isogeometric Analysis 
47 
of elements, the basis functions are not confined to one element but extend over a 
series of elements, as illustrated in Figure 3.11. This is a very important difference 
to classical finite elements because it allows higher continuities of shape functions 
over the element boundaries. 
As in the p-version of the finite elementingmethod Pilkey [41], the high-order 
nature of the basis functions generally results in higher accuracy compared to low-
order elements. In contrast to p-version elements, NURBS-elements also have high-
order continuities between elements, which is the basis for the element formulation 
presented in the next chapter. On the other hand, it means that the elements are 
interconnected and not independent of each other. The basis functions inside a knot 
span are defined by the Cox-deBoor recursion formula and depend on the neighboring 
knot spans, see Eq. 3.4. Therefore, it is not possible to define a single NURBS 
element without a complete NURBS patch. In this context, it is worth discussing the 
term elements since they are not independent, elementary parts that can be assembled 
arbitrarily to form a bigger model. 
In the implementation persepctive, these elements can be treated exactly in the 
same way as classical finite elements. The stiffness matrix, for example, is evaluated 
on element level and assembled to the global stiffness matrix. The only difference is 
the use of different shape functions. The fact that the corresponding nodes, i.e. control 
points, usually lie outside the element, is solely a consequence of the used basis 
functions and does not make any difference in the treatment of these elements in a 
finite element code. Many locking phenomena in structural analysis are a 
consequence of the low-order basis functions that cannot correctly represent the 
physical behavior Bezier [42] and Hughes [23]. Since NURBS are higher order 
functions, these locking effects can be avoided efficiently. 
The following important properties of NURBS as basis for analysis are 
summarized: 
• The basis functions fulfill the requirements of linear independence and 
partition of unity. They have a local support, depending on the polynomial degree. 
• Basis functions have higher-order continuities over element boundaries. 
CHAPTER 3: THEORETICAL FORMULATIONS 
48 
• Degrees of freedom are defined on the control points. 
• The isoparametric concept is used. 
• Rigid body motions are treated correctly (zero strains) due to the affine 
covariance property of NURBS. 
• Locking effects stemming from low-order basis functions can be precluded 
efficiently. 
3.1.10 Isogeometric Analysis versus Classical Finite elementingAnalysis 
The use of NURBS basis for geometric modelling and analysis is the 
significant difference of isogeometric analysis versus standard finite 
elementingmethod. Isogeometric analysis employs NURBS basis functions to 
construct exact geometry at all levels of discretization, while the classical families of 
interpolatory polynomial as Lagrange polynomials or Hermite polynomials are 
widely utilized in typical finite elementinganalysis. 
Major differences are listed in Table 3.1. On the other hand, isogeometric 
analysis and classical finite elementingshare many common features. For instance, 
they are both isoparametric implementations of Galerkins method, accordingly, 
isogeometric analysis inherites the computing implementation of finite 
elementingprocedure. Others are given in 
Table 3.2. 
Table 3.1 NURBS based isogeometric analysis versus classical finite element 
analysis. (Wolfgang [25]) 
Isogeometric analysis Classical finite elementing 
analysis 
- Exact geometry - Approximate geometry 
- Control points - Nodal points 
- Control variables - Nodal variables 
- Basis does not interpolate control 
points and variables 
- Basis interpolates nodal points 
and variables 
- NURBS basis - Polynomial basis 
3.2 Cotinuum-based governign equations of stability problems of inlfating beams 
49 
- High, easily controlled continuity - 
0C -continuity, always fixed 
- hpk-refinement space - hp-refinement space 
- Pointwise positive basis - Basis not necessarily positive 
- Convex hull property - No convex hull property 
- Variation diminishing in the 
presence of discontinuous data 
- Oscillatory in the presence of 
discontinuous data 
Table 3.2 Common features shared by isogeometric analysis and classical finite 
element analysis. (Wolfgang [25]) 
Isoparametric concept 
Galerkins method 
Code architecture 
Compactly supported basis 
Bandwidth of matrix 
Partition of unity 
Affine covariance 
Patch tests are satisfied 
3.2 Cotinuum-based governign equations of stability problems of 
inlfating beams 
A large number of analytical analyses related to the inflating beams and arches 
are available in literature, concerning both theoretical and experimental analysis. One 
important aspect is need to build the best adapted analytical modeling for beam 
structures. Euler-Bernoulli kinematics and the Timoshenko kinematics are widely 
used to gain the analytical solutions and to develop the formulations for inflating 
beams made of woven fabrics. Comer [3] derived a load deflection theory in the case 
of isotropic beams. Main [43] and Main [5] proposed a method for analyzing the 
inflating fabric beams with a model analogous to the shear-moment method and 
developed the theory considering orthotropic membrane model. Fichter [7] 
Analytical buckling analysisconstructed Timoshenko cylindrical inflating beams 
CHAPTER 3: THEORETICAL FORMULATIONS 
50 
made of elastic isotropic textile fabric based on energy minimization approach. 
Effects of air pressures to the load carrying capacity of the beam were taken into 
account. 
In general, the beam theoretical model is developed based on the Assumptions 
are made as follows: (i) the cross section of the inflating beam remains undeformed 
under applied load, (ii) the cross section translation and rotations are small, (iii) the 
circumferential strain is negligible. Wielgosz [10] presented analytical solutions for 
inflating plates and tubes based on Timoshenko kinematics. The work took into 
account the geometric stiffness and the residual force effect due to the internal 
pressure. They indicated that the limit load is proportional to the applied pressure and 
that deflections are inversely proportional to the material properties of the fabrics and 
to the applied pressures. In order to improve Fichter’s theory, Wielgosz [44] proposed 
a new formulation using the virtual work principle in Lagrangian form and Kirchhoff 
hypothesis with finite displacement and rotation to derive nonlinear equations of 
inflating beams. Davids [17] and Davids [18] presented nonlinear load-deflection 
response of Timoshenko inflating beams. Parametric studies have been also 
investigated in their work. Malm [19] used 3D isotropic fabric membrane finite 
element model to predict the beam load-deformation response. 
In this chapter, theoretical formualtions developed by Nguyen and his 
coleagues ([22], [52] and [130]) are employed for the buckling problems of inflating 
composite beams is presented. The obtained governing equations are then discretized 
in accordance to IGA manner in the next chapter to find the numerical solutions of 
the buckling problems. It is noted that in the previous work of Nguyen ([22] and [52]), 
the author used traditional finite element approach to solve the problem. 
3.2.1 Mathematical description of inflating beams 
In this study, we focused our work on the Timoshenko beams made from 
orthotropic material. For inflating structures, the load is applied in two stages: First, 
the beam is inflating to the pressure p, and other external forces are applied. At the 
beginning of the first step, the internal pressure is zero and the beam is in its natural 
state Figure 3.13a. The reference configuration corresponds to the end of the first 
3.2 Cotinuum-based governign equations of stability problems of inlfating beams 
51 
stage Figure 3.13b. The Green-Lagrange strain measure is used due to the 
geometrical nonlinearities. 
Figure 3.13 HOWF inflating beam: (a) in natural state and (b) in the reference 
configuration (inflating state) 
Figure 3.13 shows an inflating cylindrical beam made of an HOWF. 
0 0 0 0, , ,l R t A and 0I represent respectively the length, the external radius, the fabric 
thickness, the cross-section and the second moment of inertia around the principal 
axes of inertia Y and Z of the beam in the reference configuration which is the 
inflating configuration. 0A and 0I are given by 
0 0 02A R t = 3.22 
2
0 0
0
2
A R
I = 3.23 
where the reference dimensions 0 0,l R and 0t depend on the inflation pressure 
and the mechanical properties of the fabric Apedo [45]: 
( )0 1 2
2
lt
t
pR l
l l v
E t
 


= + − 3.24 
CHAPTER 3: THEORETICAL FORMULATIONS 
52 
( )
2
0 2
2
lt
t
pR
R R v
E t



= + − 3.25 
0
3
2
lt
t
pR
t t v
E

= + 3.26 
in which ,l R  and t 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 
inflating 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 s
L

= where 0L l= is the beam length and 
0
0
I
A
 = is the beam radius of gyration. The coefficient  takes different values 
according to the boundary conditions of the beam. 
M is a point on the current cross-section and 0G the centroid of the current 
cross-section lies on the X - axis. The beam is undergoing axial loading. Two Fichter’s 
simplifying assumptions are applied in the following: 
- The cross-section of the inflating 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. 
3.2.2 Theoretical formulation 
3.2.2.1 Kinematic relations 
The material is assumed orthotropic and the warp direction of the fabric is 
assumed to coincide with the beam axis; thus the weft yarn is circumferential. The 
model can be adapted to the case where the axes are in other directions. In this case, 
3.2 Cotinuum-based governign equations of stability problems of inlfating beams 
53 
an additional rotation may be operated to relate the orthotropic directions and the 
beam axes. This general case is not addressed here because, for an industrial purpose, 
the orthotropic principal directions coincide with the longitudinal and circumferential 
directions of the cylinder. 
With the hypotheses proposed by Fichter were applied, the displacement filed 
of an arbitrary point M(X, Y, Z) are expressed as follows: 
( )
( )
( )
( )
( ) ( )
0 0
0 0
X Y Z
Y
Z
u u X Z X Y
X
X
M u
u w X
v
    −  
= = + +    
    
u 3.27 
Where ,X Yu u and Zu are the components of the displacement at the arbitrary 
point M, whilst ( ) ( ),u X v X and ( )w X correspond to the displacements of the 
centroid 0G of the current cross-section at abscissa X, related to the base (X, Y, Z); 
( )Y X and ( )Z X are the rotations of the current section at abscissa X around both 
principal axes of inertia of the beam, respectively. Let  u denote an arbitrary virtual 
displacement from the current position of the material point M: 
( )
( )
( )
( ) ( )
0 0
0 0
Y Zu X Z X Y X
v X
w X
  
 

   − 
= + +   
   
u 3.28 
The definition of the strain at an arbitrary point as a function of the 
displacements is: 
l nl
= +E E E 3.29 
Where tE and nlE are respectively the Green-Lagrange linear and nonlinear 
strains. The nonlinear term nlE takes into account the geometrical nonlinearities. The 
strain fields depend on the displacement fields as following: 
CHAPTER 3: THEORETICAL FORMULATIONS 
54 
, ,
, ,
, ,
, , , ,
, , , ,
, , , ,
1
2
1
2
1
2
,
1 1
2 2
1 1
2 2
1 1
2 2
TX
X X
TY
Y Y
TZ
Z Z
l nl
T TX Y
X Y Y X
T TX Z
X Z Z X
T TY Z
Y Z Z Y
u
X
u
Y
u
Z
u u
Y X
u u
Z X
u u
Z Y
  
 
 
 
  
 
= =  
  + +
  
  
 ++
  
  
++ 
   
u u
u u
u u
E E
u u u u
u u u u
u u u u

 3.30 
The higher-order nonlinear terms are the product of the vectors that are defined 
as follows 
, , ,
, , ,, , ,
, , ,
, ,
X X X Y X Z
X Y ZY X Y Y Y Z
Z X Z Y Z Z
u u u
u u u
u u u
   
= = =   
   
u u u 3.31 
3.2.2.2 Constitutive equations 
In this study, the Saint Venant-Kirchhoff orthotropic material is employed. 
The energy function ( )E = E related to this case is known as the Helmholtz free-
energy function. 
To describe the behavior of the inflating beam, we define two coordinate 
systems: A local warp and weft direction coordinate system related to each point of 
the membrane coincident with the principal directions of the fabric Figure 3.14a. 
And the other is the Cartesian coordinate system attached to the beam Figure 3.14b. 
The components of the second Piola-Kirchhoff tensor S are given by the 
nonlinear Hookean stress-strain relationships 
.
o o
= + = +

S S S C E
E
 3.32 
3.2 Cotinuum-based governign equations of stability problems of inlfating beams 
55 
Figure 3.14 (a) Fabric local coordinate system, (b) Beam Cartesian coordinate 
system 
where 
- 
o
S is the inflation pressure prestressing tensor. 
- the second Piola-Kirchhoff tensor is written in the beam coordinate system 
as 
XX XY XZ
YY YZ
ZZ
S S S
S S
symmetrical S
 =
S 3.33 
- C is the fourth-order elasticity tensor expressed in the beam axes. 
In general, the inflation pressure prestressing tensor is assumed spheric and 
isotropic Wielgosz [44]. So, 
o oS=S I 3.34 
Where I is the identity second order tensor and o o
o
N
S
A
= is the prestressing 
scalar. The elasticity tensor expressed in the beam axes can be calculated from the 
local orthotropic elasticity tensor using the rotation matrix R (see Apedo [45]): 
loc
ijkl im jn kp lq mnpqC R R R R C= 3.35 
With i, j, k, m, n, p, q = 1, , 3, where 
1 0 0
0 cos sin
0 sin cos
 = −
R 3.36 
and 
CHAPTER 3: THEORETICAL FORMULATIONS 
56 
11 12
12 22
66
0
0
0 0
loc
C C
C C
C
 =
C 3.37 
The elasticity tensor in the beam axes then obtained as 
2 2
11 12 12 12
4 2 2 3
22 22 22
4 3
22 22
2 2
22
2
66 66
2
66
0 0
0 0
0 0
0 0
C c C s C csC
c C c s C c sC
s C cs C
c s C
s C csC
symmetrical c C
= 
C 3.38 
Where cosc = and sins = with ( ),Ze n = being the angle between the 
Z-axis and the normal of the membrane. The components of the elasticity tensor are 
given by 
11 12
22 66
; ;
1 1
;
1
t l tl
lt tl lt tl
t l t
lt
lt tl lt tl
E E v
C C
v v v v
E E E
C C G and
v v v v
= =
− −
= = =
−
3.2.3 Virtual work principle 
The balance equations of an inflating beam come from the virtual work 
principle (VWP). The VWP applied to the beam in its pressurized state is 
int ,
d p
ext extW W W   = +  u 3.39 
 : f . . t. ,
o o o
o o
V V V
dV dV R dA    

 = + +  S E u u u u 3.40 
where f and t are the body forces per unit volume and the traction forces per the left-
hand-side of Eq. 3.39 is formulated from the second Piola-Kirchhoff tensor S and 
the virtual Green strain E . 
The virtual Green strain tensor is written in the beam coordinate system as 
l nl
  = +E E E 3.41 
where 
3.2 Cotinuum-based governign equations of stability problems of inlfating beams 
57 
T
l l l l l l
XX YY ZZ YZ ZX XYl
E E E E E E       = E 3.42 
T
nl nl nl nl nl nl
XX YY ZZ YZ ZX XYnl
E E E E E E       = E 3.43 
with 
, , ,
, ,
,
0
0
0
l
XX X Y X Z X
l
YY
l
ZZ
l
YZ
l
XZ X Y X
l
XY X Z
E u Z Y
E
E
E
E w
E v
   



  
  
= + −
=
=
=
= +
= −
 3.44 
and 
( )
( )
( )
, , , , , ,
, , , , , ,
, , , ,
nl
XX X Y X Z X X X X
X X X Y X Z X Y X
X Y X Z X Z X
E u Z Y u v v
w w Z u Z Y
Y u Z Y
    
   
  
= + − +
+ + + −
− + −
( )
( ), , , ,
, ,
nl
YY Z Z
nl
ZZ Y Y
nl
YZ Z Y Y Z
nl
XZ Y X X Y X Z X Y
Y Y X Y Z X
E
E
E
E u u Z Y
Z Y
  
  
    
     
   
=
=
= +
= + + −
+ −
( )
, ,
, , , ,
nl
XY Z X Z Y X
X Y X Z X Z Z Z X
E u Z
s u Z Y Y
    
    
= − −
− + − +
 3.45 
The generalized resultant forces and moments, and the quantities ( )1,...,10iQ i = 
acting over the reference cross-section oA can be related to the stresses in the beam 
by 
CHAPTER 3: THEORETICAL FORMULATIONS 
58 
,
o
XX
y XY
oz XZ
A
y XX
z XX
N S
T S
dAT S
M ZS
M YS
  
=  
−  
 3.46 
2
2
, 1,...,10
o
XX
XX
XY
XZ
XX
i o
A
XY
XZ
YY
ZZ
YZ
YZS
Z S
ZS
ZS
Y S
Q dA i
YS
YS
S
S
S
− 
 −
= = 
 −
− 
 3.47 
where, N corresponds to the axial force, yT and zT to the shear force in Y and Z 
directions respectively, yM and zM to the bending moments about the Y and Z-
axis. Quantities iQ depend on the initial geometry of the cross-section: 
( )
0
0 2 2 2
11 , , , ,
1
2
XX X X X X
A
N S dA N C u u v w
= = + + + + 
( ) ( )2 2 2 212 0 11 0 , ,
1 1
4 2
Y Z Y Z Z XC A C I   

+ + + +

3.48 
( )
0
0 66 , ,
1
1
2
y XY y X Z X
A
T S dA k A C v u = = − + 3.49 
( )
0
0 66 , ,
1
1
2
z XZ z X Y X
A
T S dA k A C w u = = − + 3.50 
( )
0
, 11 , 01y XX X Y X
A
M ZS dA u C I= = + 3.51 
( )
0
, 11 , 01z XX X Z X
A
M YS dA u C I= − = + 3.52 
3.2 Cotinuum-based governign equations of stability problems of inlfating beams 
59 
and 
( )
0
2
1 0 11 0 , , 12
1
4
XX Z X Y X Z Y
A
Q YZS dA I C R C   = − = − 3.53 
( )
0
0
2 2 2 2
2 11 , , , ,
0
1
2
XX X X X X
A
N
Q Z S dA C u u v w
A
= = + + + + 
( ) ( )2 2 2 2 20 , , 12 0
1 1
3 3
8 8
Y X Z X Z YR C I   
 
+ + + +  
 
3.54 
( )
0
3 66 0 , ,
1
3
4
XY Z Y X Y Z X
A
Q ZS dA C I    = − = − 3.55 
( )
0
4 66 0 , ,
1
4
XZ Y Y X Z Z X
A
Q ZS dA C I    = = − 3.56 
( )
0
0
2 2 2 2
5 11 , , , ,
0
1
2
XX X X X X
A
N
Q Y S dA C u u v w
A
= = + + + + 
( ) ( )2 2 2 2 20 , , 12 0
1 1
3 3
8 8
Y X Z X Z YR C I   
 
+ + + +  
 
3.57 
( )
0
6 66 0 , ,
1
4
XY Z Z X Y Y X
A
Q