Finite element analysis of laminated composite plates with multi-pin joints considering friction

Compurers d; S/rucrura Vol. 55. No. 3. pp. 507-514. 1995 Copyright c 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

Pergamon

FINITE ELEMENT ANALYSIS OF LAMINATED COMPOSITE PLATES WITH MULTI-PIN JOINTS CONSIDERING FRICTION Seung Jo Kim and Jin Hee Kim Department

of Aerospace

Engineering.

Seoul National University, Republic of Korea

(Received

20 November

Gwanak-ku,

Seoul

151-742,

1993)

Abstract-In this paper the problems in multi-pin joints of laminated composite plates considering friction effects in the contact boundary are analyzed. The coulomb friction law is used and the contact constraint are handled by extended interior penalty methods. The perturbed variational principle is adopted to treat the non-differential term due to the coulomb friction, and the obtained equilibrium equation is discretized by the finite element method. The normal contact pressures and the tangential forces due to the frictional effect in the contact area are obtained by the penalty term without the increment of the number of unknowns, as in the Lagrange multiplier method. As numerical experiments, the joint problem with a single pin hole is. first, solved for various lamination angles of the plate and parameters such as clearances between pins and holes, friction coefficients and geometric factors of the plate. The computed results are compared with the existing ones by the previous workers and these agree well. The joint systems with two pin holes in series and in parallel are also analyzed in a similar manner. It is concluded that the formulation of composite joint problems considering frictional contact by utilizing the extended interior penalty method and the regularization process for the non-differentiable term is mathematically sound, physically reasonable and numerically efficient.

NOTATION

There have been a number of research works to determine the contact stress distribution around the hole in a pin-loaded plate analytically [l-7] or numerically [8-171. When joint problems are considered as contact problems without friction, many investigators have assumed a known cosinusoidal radial traction around the hole[l2, 131. Although this assumption has been shown to be a permissible approximation for isotropic plates [18], it is not adequate for anisotropic plates [7, 161. The effects of pin elasticity [7, 11, 151, friction [I, 2, 5-7, 9, IS] and clearance between the pin and the hole [l, 2, 7,9, IS, 161 have been studied as important factors. It is generally accepted that there are two distinct regions in the contact area when the frictional effect is considered. One is a non-slip region and the other is a slip region. Because of the extreme difficulties, most of the works [I, 2,5, 61 on the frictional contact problems except a few papers [7, IS] omitted the non-slip region. Multi-pin joint problems have been analyzed by several workers [2, 12-14, 161, but analyses including both slip and the non-slip regions have rarely been carried out. In the present study, bolted joint composite plates as a class of frictional contact problem are solved by utilizing a new efficient extended interior penalty formulation for the non-frictional contact problem [I 61. The perturbed variational principle is adopted to solve the equation with a non-differential function [19,20]. Next, three representative numeri-

hole diameter edge distance components of body force per unit volume distance between two pins orthonormal basis vector in R’ plate length number of contact points normal vector normal to dR total applied force average bearing stress initial contact clearance laminate thickness components of surface tractions Cartesian components of displacement u normal component of displacement u plate width trace operator normal trace operator first variation of Q penalty parameter extended penalty parameter perturbed penalty parameter nondimensional contact clearance frictional coefficient normal pressure tangential stress (rr,,) INTRODUCTION

Mechanically fastened joints in composite structures are commonly used in aerospace vehicles. Due to the anisotropic and heterogeneous nature, joint problems in composite structures are more difficult to anlayze than those in isotropic materials. 507

508

Seung Jo Kim and Jin Hee Kim

cal examples-single hole, two holes in series and in parallel in laminated composite plates-are presented. Contact clearances between pins and holes, friction coefficients, geometric factors and the load quantity are considered as design parameters for three lamination angles, [OL], [90-l and [O”/ + 45’~/90”], But the elastic deformation of the pin is not considered since the previous work [7] has shown that the pin elasticity is not as important as the other factors. THEORY

Variational principle for the frictional

contact problem

A general class of contact problems with friction are characterized by the following set of equations and inequalities: cr,,,,+f; = 0

.io=jn./.dx+[,, t,zl,ds .i(u, v) =

on To

o,,n, = t,

on rF

(1)

~lF,,l=g, (2)

if u,--s=O a,
u,=O

b, < 0,

ur = -vvor

if ]ur] = ,uJa,],

for all v E K,

(4)

where

K={vEVIY,(V)-SGO

a(u, v) =

s R

E,,,,w+,jdx,

(10)

Fn=un on rc

~a,)
rc

l+l=‘? +uT = -vuT

on

rc,

(11)

(3)

v - u)+j(u,v) -j(u,u) >.f(v - u)

v = {v = (u, , V?) . , 4) E WW)1~lW

Lyr,).

g given in

where

if Ja,]
where R is a smooth, open and bounded domain in RN with boundary an, To is a portion of dSZon which displacements are prescribed, Tc- is the portion of 82 on which tractions are prescribed and Tc is a candidate contact boundary, and v is a non-negative constant. This system of equations and inequalities describes the signorini problems which obey the coulomb friction law. Now, we adopt the appropriate assumptions to the elastic coefficients I&, and the properties off;, t,. Then the following variational principle characterizing problems (l)-(3) is obtained [l9]: 0,

=g,

Under these assumptions, the previous boundary conditions (3) can be converted as follows:

CO, CT,=0

(9)

The functional, j(. , ). which means the work by frictional force, is convex but not differentiable. Physically, eqn (4) above describes the principle of virtual work in an elastic body restrained by frictional forces. Unfortunately, as previous work pointed out [l9], no existence theory except some special cases is available for the general elastostatics problem with friction. Specially, when normal pressure is prescribed on the contact boundary, the existence of the solution has been proved [l9]. Therefore, it is assumed that a function g exists such that

in Q

u, = 0

ff"= 0,

~I~,Wllv,l ds.

J 1-c

p]a,(u)l

and on Tc if q--s

r

(8)

= 0

on r,J (9

on r,:

(6)

u, v E V

(7)

(12)

By these modifications, ciple is obtained: a(u, v - u) +j(v)

-j(u)

the final variational

2f(v

- u) for all v e V.

Extended

interior

penalty

prin-

formulation

for

(14)

non -fric-

tional normal pressure

It is also important to get proper normal pressure to solve the full frictional contact problem. To get non-frictional normal pressure which is needed for g in the expressionj(v), the frictionless contact problem should first be solved. There are several methods for solving frictionless contact problems. Considering convenience and efficiency, we use the extended interior penalty methods implemented by us [ 161. Classical interior penalty methods have the weak point that searches for optimized solutions in the neighborhood of the constrainted surfaces are difficult, since the values of the penalty term in the infeasible domain are infinity. The exterior penalty method for

Finite element

analysis of laminated composite plates

the contact problems also has disadvantages that the iteration should start at the design points in the infeasible region and the final convergent solution lies in the infeasible region. But in the extended interior penalty methods, the finite valued function is defined as the infeasible domain and the interior barrier is moved into the feasible domain such that the penal-. ized solution lies in the interior region of the feasible domain. The extended interior penalty functional has the following form: F, (v) = F(v) + tQ (v),

Now, let us consider the following Gateaux-differentiable perturbed functional ji( . ) [20]. Using the above function, &+, j?(v) has the following form:

k(v) =

F(v) = fu(v, v) -f(v). Q(V) takes

ds.

(21)

The fact that k(v) goes to j, (v) as E -+ 0 can be easily shown. Following this procedure, the variational equation characterizing the frictional contact system is obtained as: a(+,

functional

&i(v) s 1-c

(15)

where F(v): V+R is a functional defined on V, that is total potential energy in structures, and defined as

The penalized form:

509

(16)

v) + (Dj,(q),

for all v E V.

v) =f’(v)

(22)

FINITE ELEMENT APPROXIMATION

Following the usual finite element procedure, u, and v are approximated as the following piecewise polynomial ~4: and 0’:

the following u:,lR, = GJ:(x),

$I,,

= v;&(x).

(23)

Q(v) =

number which satisfies where 6’ is negative t, = -Cc”, where C > 0 and the penalty parameter L > 0. The functional Q(V) is called the inverse barrier extended interior penalty functional. Under these situations and using results of previous paper [ 161, the following variational equation is obtained,

The approximated

for all VE V,

yh = u “i

pressure

is described

on

foi,

Perturbed

K,, is defined

in the following

In these spaces, weak formulation is, u(u$, v”) + (Dji(u&

and (Dji(u:),

v”) is expressed

(25)

is estimated.

v”) =,f’(v”)

That (26)

as follows:

“-‘U;T

(iflvTl>Q

(if )vTl < C).

on f,}.

(19)

Prior to constructing the variational equation for the frictional contact problems, we remark the regularization procedure of j(v). j(v) in eqn (12) is not differentiable due to the term 1~~1,and so cannot be solved directly in finite element theory. To regularize (~~1, a function &: V+L’(Tc) is used: 7

sense:

as follows:

-t6Q(u,).

[VT/-5

(24)

(18)

variational principle

c

, E)

e = l,2,.

K,, = {v”Iv’ E V,g, y,?(v”)- s < 0 h~c =

as

uyq, = &5;(x),

,0

and constraint and contact

V,, is defined

v,8= {v”lvl’E [CO@)]“,

tll =U,(X,) a(u,,v)--Ip,,~,,(v)]=f‘(v)

space

ye-c-

if IA ‘u!!,l > t

g 1 hl;,

if )I _ +$,I ,< E,

?Tt

(20)

Dji(“uf) =

(28)

Seung Jo Kim and Jin Hee Kim

510

where k means step number in iterative calculations. Physically, 1’_ ‘II&] > t describes the slip region in the contact area and in that case, Qi( ) exerts additional external forces which push the iterative solution toward the non-slip region. 1’- ‘u”,] < i means that the solution lies in the non-slip region, and in that case Dji() effects the system stiffness. Tangential stress on can be obtained as

For termination of this procedure, relative error of normal pressure is used as a convergence criterion:

2 ]“-‘A~J;] relative error = +---. 1

I=, Numerical

(29) if ]I$,] d ?.

EXPERIMENT

An algorithm Let us consider the composite plate as anisotropic plane stress state. The solution procedure consists of the following three steps [20, 211. Step I. First, the normal pressures are estimated by solving the frictionless contact equation (18):

where

. . K’,I = a(&~,, &I,)

(31)

.f/{ =_#“(4,J,): i,j = I, 2; %,/?=I,2

,..,

N.

(32)

Here the trapezoidal rule for the integration of the penalty term is used. This non-linear algebraic equation (30) is solved by the standard NewtonRaphson scheme and relative error of displacement is used as a convergence criterion. Step 2. Using the estimated normal pressures from the step I, the frictional contact equation (26) can be now solved:

add the obtained

Three representative numerical examples are analyzed: plates with a single hole, two holes in series and two holes in parallel. The results for the three different laminated plates, [O'1, [90 ] and [0 / k45 190 I,, are presented here. The [0 ] and [90 ] plates represent extremity in degree of orthotropy. The [0 I/i45 ;90 1, laminated plate represents a plate which has the same inplane stiffness in ail directions and is referred to as a quasi-isotropic laminate. The material properties of CFRP composite are used in the computation. The frictional coefficient and contact clearances between pins and holes are adopted as design parameters in the analyses. We set the penalty parameter t = IO ” and (‘Z -10 ‘7 and perturbed penalty parameter t = IO-“‘. It is noted that the penalty parameters are selected carefully in general, but the extended interior penalty formulation is more flexible for the magnitude of the penalty parameter. To check convergence of the solutions, relative errors are used and when these reach IO-‘. iterations are terminated. Figure I depicts the coordinate systems used in the problem of a pin joint composite plate and Fig. 2

$GG ‘$Gizi

dcT in Step 2 to

P

7(v) =fv)

oT v, ds

-t

Fig.

Steps

I and 2.

P

I) Single Hole and Two Holes in series

(35)

s and repeat

the following

(I) pins are rigid; (2) laminated composite plates have symmetric stacking sequences; (3) the friction coefficient kl is constant on the contact boundary; (4) linear elasticity holds.

where J,, J, are the integral terms including the perturbed penalty term. As remarked in the previous section, J, represents the additional external force due to frictional effects and JZ has an effect on the stiffness matrix. Step 3. Finally, fin Step I,

example

For the computational experiments, general assumptions are made:

if ]I$~] > c‘

NUMERICAL

(36)

lAGI

I. The coordinate

11) Two IHoles in parallel

system used in presenting area.

the contact

Finite element

analysis

of laminated

composite

plates

Fig. 2. Description ofcontact clearance 1:1 = (Rb - $)I&.

Fig. 4. Comparison of present results with analytic given by De Jong [5] ([02 /_+45 I,, 1. = 0).

illustrates non-dimensional tween pins and holes.

tional case. In this case, [Oi / + 45”], laminates are used and i = 0 is prescribed. The result shows good agreement. The results with friction are also compared with the analytic solution by Hyer et al. [7] in Fig. 5 and shows better agreement than Yogeswaren and Reddy [IS]. The case [0’/+45”/90”], as a generally used laminated plate is also analyzed (Fig. 6). The maximum peak stress is located away from the symmetric line of plate, unlike the non-frictional case. And the distribution of normal pressure is quite different from the non-frictional case. The non-slip region occupies less than 5.6”. Applied force is 8000 N and E. = 0. I % and p = 0.2 are prescribed. The orthotropic plates, [O”] and [90”] are considered as extreme cases to see the behavior of laminated composite with different ply orientations. It is noted that normal pressure and tangential stress distribution on the contact boundary strongly depend on geometry of the plate [l6]. The results of the [O”] laminate show signs of change of tangential stress as mentioned in Ref. [22] (Fig. 7). The non-slip region reaches about 20’. The results of the [90”] laminate are shown in Fig. 8. Since the [90”] laminate is much softer in the load direction, the location of the maximum normal pressure is located at about 70

contact

clearances,

1, be-

The case of a single hole. The geometry of the single hole model is shown in Fig. 3. Half of the plate is discretized by 262 Q, (four-node linear) element with 306 nodes by using symmetry. Finer mesh is used in the vicinity of the hole. To check the validity, the results are compared with those by De Jong [5] in Fig. 4 for the non-fric-

+W/2=80+

t= 10

R10

M

-0.0

Fig. 3. Finite element model for a single hole.

.

results

/

Fig. 5. Comparison of present results with analytic results given by Hyer CI ul. [7] ([0, /&45’ 1,. i. = 0, fl = 0.2).

Sung

512

Jo Kim and Jin Hee Kim

0.4 02 -0.c -0.2

y-o.4 10 g

-0.6

2 -0.8 -1.0 -1.2

-1.1

-1.8 0

LO

20

30

10

0

20

30

*I&$;

84

70

8b

] laminate

Fig. 6. Contact stress distribution of [0‘/+45 i90 1, laminate (P = 8000N, i. = O.l%, @ =0.2).

Fig.

When the frictional effects are included, the wider contact area is observed. All contact stresses are non-dimensionalized by average bearing stress(S).

shown in Table 1 for the [0 /+45 190 1, laminate, without friction, the upper pin resists 2926 N and the lower pin resists 5074 N. When the frictional coefficient, p, is 0.2, the upper pin takes 2866 N and the lower pin takes 5134N. From this result, it can be mentioned that the presence of friction reduces the pin loading of the upper pin and increases the pin loading of the lower pin. And when the upper pin has

TM holes in series. The half of the plate with two holes in series is discretized by 568 Q, elements with 648 nodes, as shown in Fig. 9. The computed results of [0 / _t45-190’1, laminates are shown in Fig. IO with P = 8000 N, i, = 0.1% and p = 0.2. The contact area in the computation considering friction is wider than that of the frictionless case, as founded in the single pin model. In this geometry with two holes in series, the lower pin experiences more load and wider contact area than the upper pin. The non-slip region occupies less than 5.6 in contact areas in both pins. It is noted that the width and length of plate, distance between the two pins and contact clearances have significant influences on the distribution of contact pressure. And the computed contact area and distribution of contact stress are different from those with a single hole. Also, each pin experiences different pinloading. Pin loading also varies with change of frictional coefficient. As

Fig.

7. Contact stress (P =8000N.

distribution of [0 i =0.10/u, p =0.2).

8. Contact

stress

(P =8000N,

distribution i. =O.l%,

14_ W/2=80

of [90 p =0.2).

)-I

t=10

~ L= 400

] laminate Fig. 9. Finite element

model

for two holes in series

Finite element

analysis

of laminated

composite

plates

Fig. 10. Contact stress distribution of [0”/&45”/90”], laminate (P = 8000 N, i = 0. I %, p = 0.2). bigger clearance than the lower pin, the pin loading of the upper pin is reduced and vice versa. Two holes in parallel. As the final example, the composite plate with two parallel pin joints is considered. The finite element model with 546 Qr elements and 613 nodes is shown in Fig. 11. The computed results [Oc/ & 45’/90”], laminates are shown in Fig. 12 with P=8000N, p =0.2 and 1 =O.l%. The results show the similar characteristics compared with those of single hole. But due to the different boundary conditions, i.e. symmetric condition of symmetric line and stress free condition of the edge of the plate, the contact area moves toward the symmetric line of the plate in the case with parallel holes. And the point of sign change of tangential stress is positioned toward the symmetric line of the plate. The wider contact area is also observed in comparison with the non-frictional one. CONCLUSIONS

In this paper the problem of multi-pin joints of a laminated composite plate with frictional contact conditions are analyzed. Coulomb friction law is used and the contact constraints are handled by extended interior penalty methods. The perturbed variational principle is adopted to treat the non-differential term due to the coulomb friction. The computed results by our formulations are compared with previous works in the case of single hole and agree well. And the analyses of the multi-pin joint composite plates have shown the efficiency of the formulations. The geometric factors, clearances and friction play important

Fig. Il. Finite element model for two holes in parallel. roles in determining contact stress. With variation of these factors, it is found that the each pint takes a different magnitude of pin loading, and extended parametric studies on these factors may be needed for design consideration. In the case of two holes in parallel, the distributions of contact stress show somewhat different aspects of those single holes due to different boundary conditions. It is concluded that the formulation of the composite joints problems considering friction effect is physically reasonable and 0.0

0.0

-0.6 ?

1 -I.0 45 9)

Table 1. Pin loadings for [0 ‘/i45 _

Case I Case II Case 111

190 1, laminate

Frictionless

(unit = N)

in

-La

Friction

Upper

Lower

Upper

Lower

2926 3186 2762

5014 4814 5238

2866 3170 2706

5134 4830 5294

Case I, both i. = 0%; case II, upper i = 0. I %, lower i. = 0.2%; case 111, upper ; = 0.2%, lower i. = 0.1%.

-1.0

-2.5

Fig. 12. Contact stress distribution of [0 /k45 190 1, laminate (P = 8000 N. i. = 0. I %. 11 = 0.2).

Seung Jo Kim and Jin Hee Kim

514 numerically for

the

aerospace

efficient.

design

Also

the results could be useful

of composite

structures,

as used

in

industries. REFERENCES

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