Deflection Analysis of Woven Composite Planes under In-Plane Loading

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The Twelfth East Asia-Pacific Conference on Structural Engineering and Construction

Deflection Analysis of Woven Composite Planes under

In-Plane Loading

D. DERAKHSHAN

1a

, R.T. Faal

2b

1,2_{Faculty of Engineering, Zanjan University, Zanjan, Iran}

Abstract

Deflection analysis of yarns of a bi-axial woven composite plate under tensional loading is investigated in this article. A unit part of the warp yarn which is limited by two adjacent weft yarns is modeled by a curved beam. Using the Winkler theory of curved beams, the strain energy under the tensile loading is derived. Initial shape of yarns is chosen to be arc shaped. The deflection analysis is accomplished by making use of energy method. First, the external work of the horizontal (or tensile) force and also the vertical contacting force is evaluated. Second, using the variational method the governing equations and boundary conditions are derived and then solved to find out the displacement fields exactly. Finally, making use of the continuity of the deflection of two contacting yarns at the contacting point the vertical contacting force is obtained. It must be mentioned that shear forces and slipping of contacting yarns are disregarded here. The validation of the work is performed by comparison of results with the semi circle beam (yarn) which is found in some usual text books.

Keywords: (Woven composites, In-plane loading, Curved beam, Winkler theory, Yarn)

1 INTRODUCTION

Among various kinds of materials which have superior properties such as lightness, strength, corrosion resistance, thermal and electrical insulation, composite materials have magnificent situation in mechanical engineering. During few decays ago, woven composite is under consideration because of better strength, rigidity, toughness and creep resistance performance. Literature is replete with studies related to

a_{Corresponding author: Email: [email protected]} b_{Presenter: [email protected]}

Procedia Engineering 14 (2011) 2839–2847

Open access under CC BY-NC-ND license.

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applications, fabrications and numerical analysis of woven composites. Buckling, slipping and stretching of fibers of woven composite plates which play an important rule in the behavior of those were the subject of some pertinent studies. A great deal of work (for example) (Ishikawa and Chou 1982; Naik and Shembekar 1992; Duosheng et a.l. 2005) had been done in analytical manner to obtain models for elastic deformation, stiffness, strength behaviors and even stability (buckling) predictor. The majority of these analytical models had been developed by rather simple approximating assumptions so that the reliability of models may not be confirmed completely. Also, a few investigations had been presented based on the finite element models (Haan et al. 2001; Boisse et al. 2005; Iannucci 2006). Some ones introduced an elastic woven unit cell which simulates fabric woven composite including yarns and matrix. Others accomplished works related to forming and impacting of woven composites were also analyzed by finite element models. Recently, anti-plane loading (or bending) of multi-layer woven composite was taken under consideration (Li et al. 2008). Nowadays, the experimental methods are another subject of investigation of woven composites. Picture frame and bias extension tests are well-known experimental methods which have been utilized in many papers (Harrison et al. 2004). In this work a robust analytical method is used to evaluate displacement fields based on Winkler’s theory (Langhaar 1962) which develops a linear elastic curved beam formulation. The theory incorporates the hypothesis which states that area cross section remains plane and perpendicular to centroidal line before and after deformation. Also, Hook’s law associates with variational method to derive the relations of the displacements and internal forces. The work has been compared with analytical solutions which are brought in usual mechanical engineering text books.

2 DERIVATION OF GOVERNING EQUATION OF CURVED BEAM

In the woven composite plates, fabrics play an important rule in the behavior of those. Therefore initially we analyze the fabric yarn of a woven composite plate which is considered a curved beam. Analysis was done for constant curvature based on Winkler theory (Langhaar 1962). Consider an arc shaped yarn, which

u

,

v

and

U

denotes to radial and circumferential displacement components and also the radius of curvature of a point on the centroidal axis, respectively. These displacements are functions of

s

,

the arc length measured on the centroidal axis. Suppose an ordinate,

z

in a cross section of beam with area

A

and moment of inertia

I

,

measured from centroidal axis (positive outward). According to Winkler theory of curved beams, the dimensionless constant

Z

is defined by (Langhaar 1962)

) 1 ( Z A z dA A

³

U U (1)

The net axial tension of beam

³

A

dA E

N H and the bending moment

³

A

dA z E

M H about the centroidal axis of cross section which

H

designates axial strain, are as follows (Langhaar 1962)

) ( )] ( )[ / ( 2 2 u u EAZ M u u Z v u EA N cc cc c U U U U (2) where prim denotes to differentiation with respect to argument

s

.

The strain energy of beam is

ds u u Z v u EA U

³

( /2U2)[( U c)2 ( U2 cc)2] (3)

In view of Eqs. (2), the strain energy of beam may be rewritten in terms of net axial tension of beam and the bending moment of beam as

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ds Z M N M EA U

³

(1/2U2 )[( U )2 / ] (4)

We consider a composite plate under in-plane tensional loadingP which is only applied on two opposite sides of plate. Therefore this loading is applied to parallel yarns which are only in the direction of loading. The upper yarn (as a curved beam) is subjected to axial load

P

at the right end and vertical loadQwhich is applied by other yarn on the mid point of beam length shown in Fig. 1. Consider the axial load

P

and lateral loadQ(due to contacted yarn). By virtue of Eq. (3), the total potential energy of beam is readily written as

³

cc c cc c 2 / 0 2 2 2 2 / 0 2 2 2 ] ) ( ) )[( 2 / ( ] ) ( ) )[( 2 / ( ) 2 / ( 2 sin ) ( 2 cos ) ( D D I U U T U U D D D D D d u u Z v u EA d u u Z v u EA Qu Pu Pv V (5)

Where prim designates to derivation with respect toTand

I

. Because of existence of point loadQ, the

dummy variables of the first and second integrals are

T

and

I

respectively.

P

y

u

v

x

s

Q

upper yarn

lower yarn

Figure 1: Curved beam model for woven yarns under axial and lateral loads Setting GV 0leads to the following governing equations

0 0 ) 2 ( cc c cc cc cc c v u u u u Z v u (6) Integration of the second equation with respect toT gives uvc A₆Z where A₆ is constant.

Substituting this equation into the first equation of (6) results inucccc2uccu A₆ which the solution for

u

,

v

are readily given as

1 2 3 4 6

1 2 3 4 5

1 2 3 4 6

1 2 3 4 5

sin

cos

sin

cos

;

0

/ 2

cos

sin

(sin

cos )

(cos

sin )

sin

cos

sin

cos

;

0

/ 2

cos

sin

(sin

cos )

(cos

sin )

u

A

v

A

u

B

v

B

T

T D

T

T T

T

T T

T

I

I D

I

I I

I

I I

I

(7)

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The following forced and natural boundary conditions and the continuity and jump conditions are attained by lettingGV 0 / 2 0 / 2 0 / 2 0 / 2 0 / 2 0 / 2 0 / 2 / 2 / 2 / 2 / 2

(0) 0;

(0) 0

0

( )

( ) ;

( )

( ) ;

( )

( ) ;

( )

( ) ; ( )

( )

cos( / 2)( )

sin( / 2)( )

; ( )

sin( / 2)( )

cos( / 2)( )

u

v

M

at

u

v

u

S

Q

S

N

M

u

v

M

S

N

P

T D I T D I T D I T D I T D I T D I I D I D I D I D I D

T

D

c

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The equations (8) are also derived by applying equilibrium conditions for an infinitesimal arc subjects to internal momentM , axial forceNand shear forceS. Continuity of displacements

u

and

v

and rotation

ucare important conditions which must be impose to this analysis. The shear forceShas a relation with

the bending moment M as S dM

U T

d EAZ u( cuccc)

U

which helps us to attainSin terms of

coefficientsA_iandB_i, i 1,2,...,6. It is worth to mention it that there are twelve conditions associate with

twelve coefficientsA_iandB_iwhich are solved simultaneously as follows

Z Q Z K{P } B B A A KP B B B P Q K A P Q K B A A A T )]} ( Ȥ ) ( Ȥ ) 2 / sin( ) 1 ( 2 [ 5 . 0 )] ( Ȥ ) ( Ȥ ) 2 / cos( ) 1 ( 2 [ { 2 2 )] 2 / sin( 2 ) 2 / cos( [ )] 2 / cos( 2 ) 2 / sin( [ 2 2 3 1 2 1 2 1 5 1 5 4 3 4 6 6 3 2 D D D D D D D D D D D D (9)

whereK U/(4AEZ)and Ȥ1(D),Ȥ2(D)andȤ3(D)are the four dimensional vectors as

T T T 3 1 3 -} ] 2 ) 2 / cos( 3 )[ 2 / sin( 2 ], 2 ) 2 / cos( 3 )[ 2 / cos( 2 , 4 ], 2 ) 2 / [cos( 2 { ) ( Ȥ } cos , sin , 0 ), 2 / sin( 2 { ) ( Ȥ )} 2 / sin( ), 2 / cos( , 1 , 1 { ) ( Ȥ 3 2 1 D D D D D D D D D D D D D D D (10)

Substituting the above coefficients into the displacements (7), the horizontal and vertical displacements of beam (

u

and'respectively) at the end of beam,I D/2and the midpoint T D/2can be evaluated in the following forms

Z Q Z K{P , u T )]} ( Ȥ ) ( Ȥ ) 2 / sin( ) 1 ( 2 [ 5 . 0 )] ( Ȥ ) ( Ȥ ) 2 / cos( ) 1 ( 2 [ } ) ( ) {( 6 4 5 4 2 / 2 / D D D D D D D D ' _I _D D T (11) whereȤ4(D),Ȥ5(D)andȤ6(D)are two dimensional vectors as

T T T 4[ , 3 3 -2 ]} 1 ) 2 / cos( 2 ) 2 / ( cos 3 ) 2 / sin( 8 sin { ) ( Ȥ } sin , 1 ) 2 / cos( 2 ) 2 / ( cos 3 { 2 ) ( Ȥ )} 2 / cos( ), 2 / {sin( ) ( Ȥ 2 6 2 5 4 D D D D D D D D D D D D D D (12)

In the special case, Q 0 and D S, the above-mentioned displacements are simplified to

( )

u

_{T D}_{/ 2}

'

( )

_{I D}_{/ 2}

S

U

P

/ 2

EAZ

.

For the small values of U/z, the truncated series

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] ) / ( ) / ( 1 )[ / 1 ( ) /(

1 Uz | U U z U z 2 is used to evaluate the integral (1) and we arrive atZ |I/(AU2).Therefore

( )

u

_{T D}_{/ 2}

'

( )

_{I D}_{/ 2}

S

U

3

P

/ 2 .

EI

The lower yarn (weft) is only under the vertical contacting forceQand for upper yarn (warp) of course bothPandQ. Thus we may

rewrite the Eqs. (11) for the lower yarn, Fig. 1, by settingP 0 and replacingQbyQ. The vertical displacements of two yarns at the contacting point, T D/2are identical which is implies that

P Q O(D) where

2 2

1 sin 2 3cos / 2 2 cos / 2 1 2 1 sin / 2 3sin 8sin / 2

Z Z D D D D O D D D D D D ª º _¬ _¼ (13)

SubstitutingQ O(D)P into Eqs. (11) gives the displacements in terms of external forcePas follows

Z Z KP{ , u T )]} ( Ȥ ) ( Ȥ ) 2 / sin( ) 1 ( 2 )[ ( 5 . 0 )] ( Ȥ ) ( Ȥ ) 2 / cos( ) 1 ( 2 [ } ) ( ) {( 6 4 5 4 2 / 2 / D D D D D O D D D D ' _I _D D T (14) Moreover, the internal forces such as axial and shear forces and also the internal moment can be easily

derived at any points using Eqs. (7) and (9). After some manipulations, the results are

° ° ° ¯ ° ° ° ® ° ° ° ¯ ° ° ° ® ) sin sin )( ( 2 1 cos cos / 0 sin ) ( 2 1 cos / cos ) ( 2 1 sin / ) sin ) )(sin( ( 2 1 ) cos( cos / 0 ) sin( ) ( 2 1 ) cos( / ) cos( ) ( 2 1 ) sin( / 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 D I D O I D U P D I I D O I K I D O I [ D D T D O D T D U P D T D T D O D T K D T D O D T [ P M for P N p S P M for P N P S (15)

where the dimensionless terms[,KandPcan be useful to analysis of the magnifications of the internal moments and forces to external applied force. The results of Eqs. (15) are also proved by satisfying the equilibrium conditions for the beam. (see Fig. 2).

P M S N Q A N B S Q/2 M P Q/2

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3 NUMERICAL EXAMPLES AND RESULTS

As it has mentioned before, the validity of displacement relations are confirmed by comparing with special case of those available in the text books. In what follows, the displacement of single warp yarn and bi-axial woven yarns subjected to an in-plane simple tension are evaluated and compared with each other. Also, the bending moment, axial and shear forces diagrams along the centroidal axis are depicted. Specifications of steel yarns such as material properties, dimensions, number and amount of loading are given in table 1, which are used in all figures. The plot of dimensionless displacements at end('/PK)and mid-point(G /PK)versusDare shown in Fig. 3.

Table 1 Specifications of steel yarns

Yong’s modulus Yarn’s diameter lengthuWidth Number of yarns Axial load /yarn

GPa

E

207

d 0.5[mm] 1000_u1000[_mm2] ₄₀₀ _P ₁₀_[_kN_] 0 50 100 150 0 0.5 1 1.5 2 2.5 3 3.5 α

Dimensionless Horizontal Displacment (

Δ /PK) 0 50 100 150 í í í í í í 0 0.2 α

Dimensionless Vertical Displacment (

δ

/PK)

Figure 3: Left: Dimensionless end horizontal displacement, Right: Dimensionless midpoint vertical displacement The variation of the ratio of the end horizontal displacement of upper yarn (warp) in presence of lower yarn (weft) to that, in the absence of weft versus arc angle(0D80)is shown in Fig. 4. This variation is almost slow for arc angles bigger than70$.

Similar geometry and construction of warp and weft yarns imply that the vertical distance of mid-point from each support, Fig 1. equals to yarn half diameter. Also, the horizontal distance of supports of curved beam or spans of the curved beam

(

L

),

Fig. 1 is specified using the ratio of width of composite plane to the number of yarns. So, the radius U may be calculated in the terms of L and arc angleD as

)) 2 / sin( 2 /( D

U L and also in view of Fig. 1 we may easily write d 2U[1cos(D/2)]. Therefore, the

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cross section diameter of curved beam, respectively. Using the data of table 1, we obtain the span lengthL 1000/(4001) 2.506[mm],and the arc angle_D ₄₅_.₁₃$_{and also}_U ₃_.₂₆₅_[_mm_]_.

0 20 100 120 0 0.5 0.55 0.7 0.75 0.9 0.95 1 α Displacment Ratio

Figure 4: Horizontal displacement ratio when weft is present to that when weft is absent

0 10 20 30 í í í í 0 0.1 0.2 0.3 θ

Dimensionless Shear Force

0 10 20 30 0.97 0.975 0.99 0.995 1 1.005 1.01 1.015 θ

Dimensionless Axial Force

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0 5 10 15 20 25 30 35 5 í í í í í í í í 0 θ

Dimensionless Bending Moment

Figure 6: Dimensionless bending moment versus angleT

Making use of equations (15), the dimensionless displacements are evaluated which are plotted in Fig. 5. Jump in the left graph and extremums in the right are compatible with physical features of problem. The similar trend of dimensionless axial force can be seen for the dimensionless bending moment along the centroidal axis. The extremums are located at the angles nearlyT 15$andI 30$(see Fig. 6). So, it is expected that the maximum normal stress may occur at these points. Accurate solution is obtained by first differentiation of KorPwith respect toT (orI).

4 CONCLUSIONS

The main conclusions of this paper may be listed as follows:

1. The end dimensionless horizontal displacement is increased rapidly with the increasing the arc angle and reversely the midpoint dimensionless vertical displacement is reduced.

2. The variation of the ratio of the end horizontal displacement of warp in the presence of weft to that in the absence of weft is almost slow for arc angles bigger than70$and in this bond the weft leads to more stiffening or rigidity.

3. Maximum normal stress of yarn cross section occurs at the angles before and after the contacting point. (Look at Fig. 6).

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