1359-8368(95)00046-1

Composites: Part B 278 (1996) 339-349 Copyright 1:1) 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved

1359-8568/96/$15.00

Finite element analysis of bolted connections for PFRP composites

Nahla K. Hassan*, Mohamed A. Mohamedient and Sami H. Rizkallat * Structural Engineering Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt t Civil Engineering Department Faculty of Engineering, Suez Canal University, Port-Said, Egypt :j: Department of Civil Engineering, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada (Received August 1995; accepted October 1995)

Structural applications of composite laminates are increasing with the drive for high strength, lightweight component design. These applications usually require joining composites either to composites or metals. Most commonly, joints are formed using mechanical fasteners. Therefore suitable prediction methods must be developed to determine failure strength and failure modes of these bolted connections.

In this paper, three-dimensional finite element analysis is conducted on single and multi-bolted connections to determine the failure process, ultimate load and load distribution among the fasteners, taking into consideration bolt-hole contact problem in the finite element model. The strength analysis is based on Tsai-Wu tensor polynomial failure criterion which is applied to the laminate as a whole.

(Keywords: composite materials; composite laminates; bolted connections; finite element; tensor polynomial failure criterion)

INTRODUCTION

With the advent of composite materials and their wide use in highly stressed lightweight constructions, it has become necessary to develop a deep insight into the failure and strength of composite bolted joints. The strength of such joints depends on the number and spacing of bolts, edge distance, bolt clearance, load distribution among the bolts and material properties. Current design procedures for mechanical joints of orthotropic materials tend to be empirically based and exist for only simple cases. Insufficient knowledge of the stress distribution in the neighbourhood of loaded holes in anisotropic materials and their interaction with the material properties have inhibited development of adequate design theory. For this reason, many finite element procedures have been proposed for analyzing composite bolted connections. Agarwal l used finite element technique and the average stress criterion over a characteristic distance d (from the hole boundary) as the stress level to predict the failure strength of the laminate. The parameter d is considered as a material property, Garbo and Ogonowski 2 used the characteristic distance concept of Whitney and Nuismer3 to determine the strength of composite laminates with unloaded fastener holes. According to this hypothesis, they chose

a characteristic distance d away from the hole and used the stress level at this distance to compute the failure loads of different laminates by the Tsai-Hill criterion4

.

The average stress criterion considers only one stress component for predicting the strength of the laminate. In fact, at almost all the points around the hole boundary, a biaxial state of stress exists. Further, with the change in ply orientation of the laminate, the points of stress concentration also change. Tensor polynomial failure criterion has been used, because it does not require additional constant d, and all the stress components are accounted for in this criterion. Sonis used Tsai-Wu6

tensor polynomial failure criterion to predict the ultimate failure strength of single bolted joint for OO±45° lay-ups and 900 ±45°. York et at.7 predicted the net tensile strength of composite bolted joints using the Sap-V finite element computer program using a two dimension FEA employing quadratical elements with orthotropic material properties. Wang and Hanns in 1988 used a general purpose finite element program NF AP, where the plates were modeled by the use of eight noded plane elements with anisotropic material proper­ties. As for the fasteners, they proposed two-dimensional pure shear elements and undergoing pure shear strain along the distance between the middle surfaces of the two plates.

339

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

One of the problems in multi-bolt joint design is that the load is not equally distributed around the bolts, especially in joints having more than two rows of bolts. Thus there is an interaction of stress fields around the bolts which is not fully understood or determined. In this paper, a three-dimensional finite element analysis using the ANSYS program9

, was used to perform stress analysis of single and multi­bolted double shear lap connections of glass fiber reinforced plastic, taking into consideration the bolt­hole contact problem. The Tsai-Wu tensor poly­nomial has been applied as the failure criterion to predict the ultimate failure strength of the composite bolted connection. The predicted load distribution among the bolts and the predicted strains using the finite element program were compared with the experimental measured values and good agreement was achieved.

OVERVIEW ON THE EXPERIMENTAL OBSERVATIONS

The behaviour of bolted joints is highly dependent on the geometric dimensions of the connection, including the edge distance, width and pitch between the bolts, in addition to the number of bolts and the bolt pattern. The experimental investigation conducted at the University of Manitoba to study the behaviour of multi-bolted connections of five different configurations, including test results, modes of failure, and load distribution among the bolts, are included in another paper. In this investigation, two basic failure modes consisting of net tension and cleavage failure were observed for the different geometric dimensions considered in the experimental program lO

. Test results indicated that the net tension failure was characterized by sudden crack propagation and hence failure was catastrophic in nature. The cleavage failure mode on the other hand, had a much more ductile behaviour. All the connections tested showed signs of bearing mode damage in the vicinity of the bolt hole after failure, where damage occurred in the laminate area adjacent to the loaded half of the hole. Some of the important observations and conclusions derived from the experimental program are summarized as follows:

(1) The ultimate load capacity of the five different configuratidns of the multi-bolted connections, shown in Figure 1, increased by increasing the geometric dimensions of the member. However, it was found that increasing the edge distance (e) to hole diameter (d) ratio (eld) beyond 5 or the side distance (s) to pitch (p) ratio (sip) beyond 1.2 had an insignificant effect on the ultimate load capacity of the connection.

(2) The load-strain relationship for all types of con­nection is linear up to failure. For connections with one row of bolts, the load is equally shared among

I I - ._ .. ~ .. _ .. - --r----T- .'

I _I: __ ~' ___ _

____ $1-2 ___ _ I

Joint A

I I I - -""T--r--,- •.

I I I I P , P I

1-1++1

3 iii I I~·$l-E;>t-If

! !2! If

JointD

! p ! ,+----+0, I I

JointB

Figure 1 Connection configuration

- -~··--I- .. 4 I 3 I I

I-t·~-?I- . --~.--$r:_ ~2

JointE

I - ----r---- .. I

4 I I 1---f-------,tE;>----!"f p

----?+i---f e

Joint C

the bolts. For connections with more than one row of bolts, the bearing forces of the fasteners are not evenly shared as measured by the gauge readings.

(3) The increase in the ultimate capacity of the con­nection is not directly proportional to the increase of the number of bolts. The effect was more significant when increasing the number of bolts in a column than when increasing the number of bolts in a row.

(4) Variation of the bolt pattern has a minor effect on the gross efficiency of the connection, when com­paring connections with the bolt in a row to those with the bolts in a column (same number of bolts in each connection).

FINITE ELEMENT ANALYSIS

Problem definition

The single bolted connection at three fiber orienta­tions was investigated analytically using the ANSYS program. In addition, one specimen of each type of the two different joint configurations at 0°, 90° and 45° fiber orientation, previously tested experimentally, were studied as given in Table I.

The fiber reinforced composite material used in this investigation is Extren flat sheet series 500, a pultruded glass fiber sheet produced by Morrison Fiber Glass Company (MMFG). The composite material is ortho­tropic, consisting of symmetrically stacked alternating layers of identically oriented unidirectional E-glass rovings and randomly oriented E-glass continuous strand mat. The mat provides multidirectional strength properties, whereas the continuous rovings provide strength in the longitudinal (pultruded) direction. The matrix consists ofisophthalic polyester plastic, which is a general duty resin that is resistant to most acidic attacks, and the fiber content is approximately 40%. The material

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

Table 1 Finite element specimens

Edge Fiber Specimen Width distance angle

. Joint pattern designation (in) (in) (degrees)

Single OSI5 6 1.5 0

45S15 6 1.5 45

90S15 6 1.5 90

A OA7 8 1.5 o

OA9 8 4.0 o

45A3 8 4.0 45

90A6 8 4.0 90

B OB3 8 4.0 o

45B3 8 4.0 45

90B3 8 4.0 90

Thickness (I) = 0.5 in, Pitch (p) = 3.25 in.

Table 2 Material properties

Thickness (mm)

Fiber angle (deg.)

Tensile modulus (GPa)

Elongation (%)

Tensile strength (MPa)

Compressive strength (MPa)

Shear strength (MPa)

12.7 o 90

12.7 10.5

Shear modulus = 4.3 GPa; Major Poisson's ratio = 0.29.

\.6 1.4

and the strength properties of the specimens studied are shown in Table 2.

Due to the high percentage of the randomly distributed fiber plies, the laminate is analyzed as one transversly isotropic layer with the assumption that the material is elastic and nonlinear.

Displacement model

In general, the distribution of the displacement over the domain of the element and the formulation of the displacement model is achieved by using either a simple polynomial model in terms of generalized co-ordinates or an interpolation model. It is possible to avoid computational work by selecting an interpolation function as the basis for the displacement model. This interpolation function, also known as a shape function, has a unit value at one nodal point and zero value at all other nodal points. Thus, with inter­polation models the analysis will include directly the nodal displacements rather than the generalized displacements. In this study the three-dimensional shell

166 110

175 145

27.8 24.0

element's shape functions for the corner nodes and the mid-side nodes, respectively, are:

Ni = * (1 + ~U(l + rrrli)(~~i + rrrli - 1)

N, =!(1 - e)(l + rrrI,) at~, = 0

N, =!(1 -1]2)(1 + ~~/) at 1]/ = 0

(1)

where ~, 1], are the local co-ordinates in the center of the element.

The element displacement field {u} e can be expressed in terms of nodal translations and rotations and the shape function {N;} e as:

(u)ie) ~ t,{N;)('){:}

+ t, (N,)(') (~; r :; lat

(2)

341

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

where Ui, Vi, Wi are the global nodal displacements, exi ,

eYh ezi are global nodal rotations, ti is the nodal thickness, ( is the local co-ordinate in the through thickness direction, and, at, a~, at are the direction cosines of the u, V and W directions with respect to the nodal co-ordinate system at node i for the vector a.

In the isoparametric formulation the co-ordinates of a point within the element are obtained by applying the element shape functions to the nodal co-ordinates:

n

X(e) = 2:)Ni} (e) {XJ(e)

i=1

n y(e) = i)NJ(e){YJ(e) (3)

i=1

n Z(e) = 2:)N;}(e){Z;}(e)

i=1

where for a node (i) of element (e), {Ni}(e) are the matrix shape functions and they are the same shape functions use~ in the element displacement field, (n) is the number of nodes per element, and {XJ(e), {YJ(e), {ZJ(e) are the nodal co-ordinates.

Stress-strain relationship

The generalized Hookes law relating stresses and strains for a linear elastic anisotropic and homogenous material (triclinic symmetry) is:

[0'] = [D][£J, which on expansion yields

Ux Dll D12 DI6 Ex

Uy D21 D22 D26

Ey

UZ D31 D32 D36

Ez (4)

Txy rxy

Tyz ryz

Txz D61 D62 D66

rzx

where { u}, {E} are the stress and strain vectors, respectively, also:

[£] = [S][O'] (5)

The matrices [D] and [S] are the stiffness modulus and the compliance matrix, respectively.

However, a complete constitutive description of a general anisotropic solid necessitates the experimental evaluation of 21 elastic independent constants out of the 36 constants If. If any material symmetry exists, the number of constants will reduce. Therefore, for a monoclinic material the 1-2 plane or z = 0 is the plane of symmetry and the number of independent constants reduces to 13 out of 20 non-zero constants. As the level of material symmetry increases, the number of independent constants continues to reduce. If symmetry exists in three orthogonal planes, as is the case for orthotropic material, the number of independent constants becomes 9 out of 12 non-zero constants, provided that the planes of symmetry coincide with the reference co-ordinate system. But if the

axes of the reference co-ordinate system are not coincident with the symmetry planes, the number of non-zero constants will be the same as for the triclinic material. The next level of material symmetry is the transversely isotropic material, which has 5 independent constants out of 12 non-zero components.

Since the glass fiber reinforced plastic sheets used in this investigation are formed of alternative layers of unidirectional E-glass fiber rovings and randomly oriented E-glass continuous strand mat, the material is considered to be transversely isotropic. The stress-strain relationship is defined as:

[O'](e) = [D](e)[£](e) (6)

Therefore, within each element the relationship is:

[O'](e) = [D](e) [B](e) [q](e) (7)

where for transversely isotropic material, the material stiffness matrix [D] is:

Dll DI2 D13 0

D21 D22 D23 0

D31 D32 D33 0 [D] =

0 0 0 (D22 - D23 )

2 0 0 0 0

0 0 0 0

where D;j are the stiffness coefficients:

Dll = (1 - Vi3)AEI

D22 = D33 = (1 - V2I VI2)AE2

DI2 = Dl3 = V21 (1 + v23)AE2

D23 = (V23 + V21 vu)AE2

E2 D44 = (1 - v23 - 2v12v21)A 2

Dss = D66 = GI2

1

0

0

0

0

Dss

0

A=-,----__ ------(1 + v23)(1 - V23 - 2v21vd

0

0

0

0

0

D66

(8)

(9)

where EI is the section modulus in the direction of the fibers, E2 is the section modulus in the transversal direction to the fibers, and G12 is the shear modulus.

Failure criteria

For the purpose of material characterization and design, a simple strength criterion for composites is essential. Strength covers many aspects associated with the failures of materials, such as fracture, fatigue and creep, under quasi-static or dynamic loading, exposed to corrosive environment and subjected to uniaxial or multi­axial stresses in two- or three-dimensional configurations. Failure of composites is further complicated by a multitude of independent and interacting mechanisms, which include fiber breakage, micro-buckling, delamina­tion and crack propagation. In addition to the different

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

tensile and compressive strengths, the strengths along the fibers are different from those in the transversal direction. Therefore, there are four uniaxial failure strengths X, X', Yand y' for a composite material, in addition to the shear strength which is also independent. This makes a total of five strength characteristics. The objective of the failure criterion is to select an envelope that will define the strength of an orthotropic ply under combined stresses, since all plies in a laminate are under combined stresses. The key issue in comparing the merits of Tsai-Wu failure criterion6 to other criteria is the interaction among the stress components, which are independent material properties in the Tsai-Wu criterion, while they are fixed (not independent) in Hill's criterion 12. In the maximum stress or the maximum strain criterion, six simultaneous equations are required and interactions are not admissible ll

.

The basic assumption of the strength criterion is that there exists a failure surface in the stress space in the following scalar form:

(10)

where the contracted notation is used and i, j and k = 1-6. F; and F;j are strength tensors of the second and fourth rank, respectively. The linear term (0";) takes into account internal stresses which describe the difference between positive and negative stress-induced failures. The quadratic term (0"; O"j) defines an ellipsoid in the stress space. Higher order terms (Fijd, (O";O"jO"d are ignored to avoid open-ended failure surfaces. Certain stability conditions are incorporated into the strength tensors, where the magnitude of the interaction terms F;j are constrained by the following inequality:

F;;FiJ-F;]~O (11)

where i, j = 1, ... ,6. Geometrically this inequality insures that the failure

surface will intercept each stress axis and the shape of the surface will be ellipsoidal, and not open-ended like a hyperboloid. When expanding equation (11) in a matrix form and for a triclinic material in three-dimensional space, the F;, F;j components would be:

Fl

F2

F3 F;= also

F4

F5

F6

Fl F12 F13 F14 Fl5 F16

F22 F23 F24 F25 F26

F33 F34 F35 F36 ( 12)

(symmetric) F44 F45 F46

F55 FS6

F66

If a material has some form of symmetry, some of the interaction terms will vanish. For specially orthotropic materials, the terms F4, F5, F6 will vanish. Also, the coupling between normal and shear strengths will vanish, i.e. FI6 = F45 = F56 = F46 =. zero. But the coupling between the normal strengths will remain.

For transversely isotropic material with the 2-3 plane as the isotropic plane, F2 = F3, F12 = F13 , F22 = F33 , and F55 = F66 , also F44 = 2(F22 - F23)' Thus, for transversely isotropic material the F; and F;j components are:

FI

F2

F2 F;=

0 also

0

0

Fll F12 FI2 0 0 0

F22 F23 0 0 0

F22 0 0 0

(symmetric) 2(F22 - F23 ) 0 0

F66 0

( 13)

Therefore, the number of independent components reduces to 2 and 5 for F;, Fij' respectively.

Connection modelling

The 8-node layered shell element was used to idealize the GFRP composite plate and the high strength steel bolts used for the bolted connection. This element (stiff 99) has six degrees of freedom at each node: translations in the nodal X, Y, Z directions and rotation about the nodal X, Y, Z axes. The geometry, node locations and the co-ordinate system for this element are shown in Figure 2(a). The element is defined by 8 nodes, average or corner layer thickness, layer material direction angles and orthotropic material properties.

There are several possible ways to treat the bolt/hole contact problem in the finite element model. One is to assume a certain contact pressure distribution acting along the boundaries of loaded holes, for example a cosine distribution; another is to assume radial displacements equal to zero on the hole boundary, which yields a distributed contact reaction to the applied load. Performing a complete analysis is a third possibility, which obviously is the most accurate. In this investigation, contact stresses and stresses in the vicinity of the hole are calculated with account taken of the contact problem. The clearance between the hole and the high strength steel bolt was represented by a three­dimensional gap element. The contact element (stiff 55)

343

Finite element analysis of bolted connections for PFRP composites: Nahla K Hassan et al.

(a)

.,-X,Ui e

Xi

(b)

K

T

_\ yGap J

~--------------------------- y

(c)

! =-------------------------- y

Figure 2 Element types

Force

represents two surfaces which may maintain or break physical contact and may slide relative to each other. The geometry and node locations are given in Figure 2(b). The element is defined by two nodes and the interface is assumed to be perpendicular to the I -J line as shown in Figure 2(b).

Since at the start position these gap elements are in open status, a three-dimensional spring element of very small stiffness is associated with each gap element. The spring element (stiff 14) has longitudinal or torsional capability in one-, two- or three-dimensional applica­tions. The ge~metry and node locations for this element are shown in Figure 2(c). The definition of all these types of elements is given in the ANSYS manua19

.

. As for the boundary conditions shown in Figure 3, the area of the steel bolts is held fixed due to the high rigidity of the bolt compared to the GFRP plate (nearly 16 times). The bottom and the left edges of the plate are free. The load was applied in terms of pressure per length (width of the specimen) on the top edge of the plate. These boundary conditions simulate exactly the con­ditions during experimental testing of the bolted con­nection using the MTS machine. By taking advantage of

344

I C.L.

2

4

4

3 i C.L.

Figure 3 Boundary conditions

Boundary Conditions

: Applied load 2,3,5: Free 6 : Fixed 4 : X direction simply

supported

Bolt

the symmetry about the Y-axis, only half of the plate is analyzed. As a result, the model represents a rigid frictionless bolt and the contact surface between the bolt and the laminate is semicircular.

Due to the high percentage of the randomly distributed fiber plies, the laminate is analyzed as one transversely isotropic layer with the assumption that the material is elastic and nonlinear.

The stress components, obtained by the finite element analysis at the centroid of each element around the bolt hole, are used to predict the maximum allowable stress in the laminate through the use of the tensor polynomial failure criterion. For three-dimensional analysis, the Tsai-Wu polynomial used in the program is:

where ¢> is the output of the Tsai-Wu failure criterion, X, X' are the tension and compression failure stresses in the x-direction, Y, y' are the tension and compression failure stresses in the y­direction, Z, Z' are the tension and compression failure stresses in the z-direction, Fxy is the X - Y coupling

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

" each element at a certain loading case and the values of X, X', Y, y', S, Fxy are input data obtained from lab tests.

Figure 4 Mesh generation for joint type B

coefficient, Fxz is the X -Z coupling coefficient, and Fyz is the Y -Z coupling coefficient.

Since in the problem under investigation pure tension is applied to the plate, the value of the output stresses U xz = Uyz = Uz = 0, most of terms vanish and equation (14) becomes:

( 15)

where the values of u" U y' u xy are the stresses obtained in

RESULTS AND DISCUSSION

The mesh used for the multi-bolted connection type B is shown in Figure 4. Four element sizes are chosen to ensure that a dense mesh is obtained in the vicinity of the bolt holes and a less dense one at the composite plate free edges.

As the load is gradually applied to the specimen the lower gap elements close and the upper ones open. These gap elements are fictitious in that they offer no resistance to the relative motions of two adjacent continua, except to prevent those two regions from overlapping under the applied load. Therefore, the adjacent regions under the applied load are free to deform such that a portion of their boundaries may be in contact, but may not cross. This is achieved mainly by assigning to these gap elements, which are fully closed, a modulus sufficiently large to allow contact of the neighbouring boundaries while preventing overlapping of these boundaries. Maximum displacements in the direction of fibers, Uy ,

occurred at the top of the plate where the load is applied. Also, maximum displacement in the transverse direction occurred at the elements near the edge of the plate at the section of the most stressed bolt, which is the upper one. This verifies the test results which indicate that the inner most bolt is the most loaded bolt. The analytical load­displacement relationship is parallel to the measured

501~------------------------------------------------------------------------------------~

Max. Load = 214.12 kN w=8", e=4" Fibres @ 90 deg.

o 2 3

,

-:It-+---------I 'w ' I Joint B

[ - Experimental···· .... ·· Fimte Element J

456

Displacement (mm) 7 8 9

Figure 5 Experimental versus finite element results for load-displacement at 90' fiber orientation (pattern B)

10

345

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

...... c: Q)

E: 1. Q) U (0

Q .~ Q

o.

Single Bolt

-Finite Element -Experimental

Joint A Joint B l~ ___ FI_·b_re_s __ @ __ 4_5_d_e_g_. __ ~)

Joint Types

Figure 6 Experimental versus finite element results for load-displacement at 45° fiber orientation

data when the slip is ignored, as shown in Figure 5 for connection type B at 90° fiber orientation, and similar results were also obtained for all connections considered in Table 1. Results shown in Figure 6 show good agreement between the experimental and the values predicted by the finite element program, at ultimate load, for all of the connections considered in Table 1 for the 45° fiber orientation, and the same results were obtained for 0° and 90° fibre orientation. Once the bolts slipped into bearing, tensile stresses and bearing stresses are induced in the vicinity of the bolt holes. Figure 7 shows the developed net tensile and bearing stresses at ultimate load for the single bolted connection. This figure shows

Figure 7 Stresses in the fiber direction (Sy) for single bolted connection

that the magnitudes of both kinds of stresses are very high at the hole boundary, but then decrease rapidly away from the bolt hole at a distance equal to half the hole diameter. The developed net and bearing stresses, at ultimate load, are in good agreement with the calculated values based on experimental results, for the three fibre orientations: 0°, 90°, 45°, as shown in Figures 8 and 9, respectively. These stresses are computed at the most stressed bolt of the connection, which is located at the upper row of bolts.

Analytical results showed that the load distribution among the bolts, for connections having two bolts in a column, is not equal, which confirms the measured experimental results.

For connections with two rows of bolts as type 'A', Figure IO(a) shows that the average net or membrane stress developed at ultimate load at the upper bolt is 60 MPa, while Figure IO(b) indicates that the membrane stress developed at the section of the lower bolt is 24 MPa, which is 40% of the total tensile stress developed at the upper bolt. This indicates that 60% of the applied load is resisted as bearing force by the upper bolt.

The unequal load distribution among the bolts is verified by the value of the predicted bearing stress, by the ANSYS program, for connection type 'A' having two bolts in a column, where bearing stress (membrane) of the upper bolt is 1.5 times that of the lower one. The same load distributions among the bolts were also obtained for the connection type 'A' with the fibers at 90° and 45° with respect to the applied load.

The behaviour of the fiber composite connection type

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

10'~------------------------------------------------------------------------,

8 •..........•...• ~ ...................... . ............ ~ ............. =r e ~I1Q-- [ Fibres @ 0 deg. ]

6

w I I w

Joint B Joint A

Fibres @ 90 deg.

4

2 I±1=r' I w I

Experimental

Finite Element

Single Bolt

Single A B Joint Types

Figure 8 Experimental versus finite element results for net tensile stress

'A' was compared to that of a similar connection made of steel. The finite element results showed that for the steel connection 50% of the net stresses were developed at the section of the lower bolt, thus indicating an equal bearing force distribution between the two bolts.

The longitudinal strains predicted by the finite element program along the net section of the most stressed row of bolts, which is the upper one, were compared to the experimental values measured from the strain gauge readings at failure load, as shown in Figure II. Good agreement was obtained for the two different joint configurations with the fibers at 0° and 45° with respect to the applied load.

FEM plots of the failed elements using Tsai-Wu failure criteria are shown in Figure 12, for connection type 'A' at 0° fibre orientation. This figure shows that the mode of failure is a mix between bearing under the bolt and longitudinal tension in the plate at the bolt sides. The total region in the vicinity of the bolt has failed, but it did not give any indication of the exact mode of failure, whether cleavage failure or net tension failure. The ultimate load capacity of the connection was predicted through running the ANSYS routine a number of times at several load increments and then checking failure using Tsai-Wu criteria. The predicted failure load by the finite element program was 20% less than the experimental failure load, and failure was indicated by

failure of a number of elements beyond a distance equal to half the diameter of the bolt, which is beyond the area under the washer. This behaviour is similar to the failure mechanism observed during testing, where as soon as the crack appeared beyond the washer, it propagated quickly followed by sudden failure.

CONCLUSIONS

The following conclusions can be drawn from the experimental studies presented in this paper.

(1) The predicted strains and load-displacement relationship of the specimens analyzed with the fiber orientation at 0°, 45° and 90° with respect to the load, by the finite element analysis, were in good agreement with the measured experimental values, thus indicating the validation of the proposed finite element model.

(2) Analytical results show that the magnitudes of the net tensile stresses are very high near the hole boundary, but then decrease rapidly away from the bolt hole at a distance equal to the hole diameter. This confirms the inefficiency of very wide joints.

(3) The finite element analysis verified the uneven distri­bution of bearing forces among the bolts of the FRP

~47

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

501~--------------------------------------------------------------------~

1 !

40 ................ ~ ...................... .

... _-+.=t · w I

-~i?--··················· I 'w : I Joint B

Joint A

10 m=t. Experimental

I w I Finite Element

Single Bolt

Single A B Joint Types

Figure 9 Experimental versus finite element results for bearing stress

A .. _ .. _ .. -

I .-~.­

I --t---i

B

p

e

Figure 10 Net stress at (a) upper bolt and (b) lower bolt of joint type A

Single

Figure 11 Measured and predicted longitudinal strains for joint type A at 45° fiber

connections having more than one row of bolts, when compared to steel connections of similar geometry.

(4) The developed net and bearing stresses from the finite element analysis are in good agreement with the calculated values based on experimental results for the three fiber orientations studied in this investigation.

(5) Tsai-Wu failure criterion used in this analytical study showed a total region of failure in the vicinity of the bolts but it did not give any indication of the exact mode of failure of the connection, whether net tension or cleavage failure.

(6) The finite element program predicted the failure load of the connections with an average of 15% difference less than the measured experimental values.

Figure 12 Failure criterion values for joint type A at 0° fiber

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

REFERENCES

Agarwal, B.L. Static strength prediction of bolted joints in compositvematerial. AIAAJ.1980, 18(11),1371-1375

2 Garbo, S.P. and Ogonowski, 1.M. Effect of variances and manufacturing tolerances on the design strength and life of mechanically fastened composite joints, AFFDL-TR-78-179, Air Force Flight Dynamics Laboratory, 1978

3 Whitney, J.M. and Nuismer, R.1. Stress fracture criteria for laminated in composites containing stress concentrations. J. Compos. Mater. 1974,8,254-265

4 Azzi, V.D. and Tsai, S.W. Anisotropic strength of composites. Experimental Mechanics 1965, 5, 283-288

5 Soni, S.R. 'Failure Analysis of Composite Laminates with a Fastener Hole, Joining of Composite Materials', ASTM STP 749 (Ed. K.T. Edward), American Society for Testing and Materials, Philadelphia, 1981, pp. 145-164

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7

8

9

10

11

12

Tsai, S.W. and Wu, E.M. A general theory of strength for anisotropic materials. J. Compos. Mater. 1971,5,58-80 York, J.L., Wilson, D.W. and Pipes, R.B. Analysis of the net tension failure mode in composite bolted joints. J. Reinforced Plastics & Composites 1982,1, 141-152 Wang, S. and Hann, Y. Finite element analysis for load distribu­tion of multi-fastener joints. J. Compos. Mater. 1988, 22, 124-135 . 'ANSYS Engineering Analysis System', Theoretical Manual, Swanson Analysis Systems, Inc., Houston, PA, 1993 Hassan, N.K. 'Multi-bolted connections for fiber reinforced plastic structural members', Ph.D Thesis, Structural Division, Ain-Shams University, Cairo, Egypt, 1995 Tsai, S.W. 'Composites Design', 3rd Edn. Think Composites, Dayton, Paris and Tokyo, 1987 Hill, R. The Mathematical Theory of Plasticity', Oxford University Press, London, 1950

•

•

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

REFERENCES

Agarwal, B.1. Static strength prediction of bolted joints in compositve material. AIAA J. 1980,18 (11),1371-1375

2 Garbo, S.P. and Ogonowski, J.M. Effect of variances and manufacturing tolerances on the design strength and life of mechanically fastened composite joints, AFFDL-TR-78-179, Air Force Flight Dynamics Laboratory, 1978

3 Whitney, J.M. and Nuismer, R.J. Stress fracture criteria for laminated in composites containing stress concentrations. J. Compos. Mater. 1974,8,254-265

4 Azzi, V.D. and Tsai, S.W. Anisotropic strength of composites. Experimental Mechanics 1965,5, 283-288

5 Soni, S.R. 'Failure Analysis of Composite Laminates with a Fastener Hole, Joining of Composite Materials', ASTM STP 749 (Ed. K.T. Edward), American Society for Testing and Materials, Philadelphia, 1981, pp. 145-164

6

7

8

9

10

II

12

Tsai, S.W. and Wu, E.M. A general theory of strength for anisotropic materials. J. Compos. Mater. 1971,5,58-80 York, J.1., Wilson, D.W. and Pipes, R.B. Analysis of the net tension failure mode in composite bolted joints. J. Reinforced Plastics & Composites 1982, 1,141-152 Wang, S. and Hann, Y. Finite element analysis for load distribu­tion of multi-fastener joints. J. Compos. Mater. 1988, 22, 124-135 . 'ANSYS Engineering Analysis System', Theoretical Manual, Swanson Analysis Systems, Inc., Houston, PA, 1993 Hassan, N.K. 'Multi-bolted connections for fiber reinforced plastic structural members', Ph.D Thesis, Structural Division, Ain-Shams University, Cairo, Egypt, 1995 Tsai, S.W. 'Composites Design', 3rd Edn. Think Composites, Dayton, Paris and Tokyo, 1987 Hill, R. 'The Mathematical Theory of Plasticity', Oxford University Press, London, 1950

349

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

50~---------------------------------------------------------------------------.

40

--~ '- 30 CI) CI)

~ ..... CI)

0> c:: .~ 20 <tS

~

10

~I--1-t---I : w : I Joint B

ill=+, I w I

Single Bolt

Single A

Figure 9 Experimental versus finite element results for bearing stress

A .. __ ._--

B I .-~.

I p .-t-.

e

Figure 10 Net stress at (a) upper bolt and (b) lower bolt of joint type A

Single

Figure 11 Measured and predicted longitudinal strains for joint type A at 45° fiber

connections having more than one row of bolts, when compared to steel connections of similar geometry.

(4) The developed net and bearing stresses from the finite element analysis are in good agreement with the calculated values based on experimental results for the three fiber orientations studied in this investigation.

................ $ ...................... . Fibres @ 0 deg. --1=+ '

w I Joint A

Experimental

Finite Element

B Joint Types

(5) Tsai-Wu failure criterion used in this analytical study showed a total region of failure in the vicinity of the bolts but it did not give any indication of the exact mode of failure of the connection, whether net tension or cleavage failure.

(6) The finite element program predicted the failure load of the connections with an average of 15% difference less than the measured experimental values.

Figure 12 Failure criterion values for joint type A at 0° fiber

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

10~--------------------------------------------------______________________ ,

,

8

~I-~-t-················t·······················

·············t·············=+ e

( Fibres @ 0 deg.

I : w ' I Joint B w I

6 Joint A

Fibres @ 90 deg.

4

I±J=+. I w I

2 Experimental

Finite Element

Single Bolt

Single A B Joint Types

Figure 8 Experimental versus finite element results for net tensile stress

'A' was compared to that of a similar connection made of steel. The finite element results showed that for the steel connection 50% of the net stresses were developed at the section of the lower bolt, thus indicating an equal bearing force distribution between the two bolts.

The longitudinal strains predicted by the finite element program along the net section of the most stressed row of bolts, which is the upper one, were compared to the experimental values measured from the strain gauge readings at failure load, as shown in Figure 11. Good agreement was obtained for the two different joint configurations with the fibers at 0° and 45° with respect to the applied load.

FEM plots of the failed elements using Tsai-Wu failure criteria are shown in Figure 12, for connection type 'A' at 0° fibre orientation. This figure shows that the mode of failure is a mix between bearing under the bolt and longitudinal tension in the plate at the bolt sides. The total region in the vicinity of the bolt has failed, but it did not give any indication of the exact mode of failure, whether cleavage failure or net tension failure. The ultimate load capacity of the connection was predicted through running the ANSYS routine a number of times at several load increments and then checking failure using Tsai-Wu criteria. The predicted failure load by the finite element program was 20% less than the experimental failure load, and failure was indicated by

failure of a number of elements beyond a distance equal to half the diameter of the bolt, which is beyond the area under the washer. This behaviour is similar to the failure mechanism observed during testing, where as soon as the crack appeared beyond the washer, it propagated quickly followed by sudden failure.

CONCLUSIONS

The following conclusions can be drawn from the experimental studies presented in this paper.

(1) The predicted strains and load-displacement relationship of the specimens analyzed with the fiber orientation at 0°, 45° and 90° with respect to the load, by the finite element analysis, were in good agreement with the measured experimental values, thus indicating the validation of the proposed finite element model.

(2) Analytical results show that the magnitudes of the net tensile stresses are very high near the hole boundary, but then decrease rapidly away from the bolt hole at a distance equal to the hole diameter. This confirms the inefficiency of very wide joints.

(3) The finite element analysis verified the uneven distri­bution of bearing forces among the bolts of the FRP

347

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

-Finite Element ., Experimental

ewr L-.. ....-;-O_ .. __ O.;.-. o0-.J0

l~ __ F_i_b_re_s_@ __ 4_5 __ de_g_. __ ~)

...... c:: (])

E 1 (]) II Ctl Q .:2 Cl

Single 80lt Joint A Joint 8

Joint Types

Figure 6 Experimental versus finite element results for load-displacement at 45' fiber orientation

data when the slip is ignored, as shown in Figure 5 for connection type B at 90 0 fiber orientation, and similar results were also obtained for all connections considered in Table 1. Results shown in Figure 6 show good agreement between the experimental and the values predicted by the finite element program, at ultimate load, for all of the connections considered in Table 1 for the 45° fiber orientation, and the same results were obtained for 0° and 90 0 fibre orientation. Once the bolts slipped into bearing, tensile stresses and bearing stresses are induced in the vicinity of the bolt holes. Figure 7 shows the developed net tensile and bearing stresses at ultimate load for the single bolted connection. This figure shows

Figure 7 Stresses in the fiber direction (Sy) for single bolted connection

346

that the magnitudes of both kinds of stresses are very high at the hole boundary, but then decrease rapidly away from the bolt hole at a distance equal to half the hole diameter. The developed net and bearing stresses, at ultimate load, are in good agreement with the calculated values based on experimental results, for the three fibre orientations: 0°, 90°, 45°, as shown in Figures 8 and 9, respectively. These stresses are computed at the most stressed bolt of the connection, which is located at the upper row of bolts.

Analytical results showed that the load distribution among the bolts, for connections having two bolts in a column, is not equal, which confirms the measured experimental results.

For connections with two rows of bolts as type 'A', Figure lO(a) shows that the average net or membrane stress developed at ultimate load at the upper bolt is 60 MPa, while Figure lO(b) indicates that the membrane stress developed at the section of the lower bolt is 24 MPa, which is 40% of the total tensile stress developed at the upper bolt. This indicates that 60% of the applied load is resisted as bearing force by the upper bolt.

The unequal load distribution among the bolts is verified by the value of the predicted bearing stress, by the ANSYS program, for connection type 'A' having two bolts in a column, where bearing stress (membrane) of the upper bolt is 1.5 times that of the lower one. The same load distributions among the bolts were also obtained for the connection type 'A' with the fibers at 90° and 45° with respect to the applied load.

The behaviour of the fiber composite connection type

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

Figure 4 Mesh generation for joint type B

coefficient, Fxz is the X -Z coupling coefficient, and Fyz is the Y -Z coupling coefficient.

Since in the problem under investigation pure tension is applied to the plate, the value of the output stresses (Jxz = (Jyz = (Jz = 0, most of terms vanish and equation (14) becomes:

(II ) ( II) (J; X+X' (Jx+ y+Yi (Jy-xx'

( 15)

where the values of (Jx, (Jy, (Jxy are the stresses obtained in

each element at a certain loading case and the values of X, X', Y, y', S, Fxy are input data obtained from lab tests.

RESULTS AND DISCUSSION

The mesh used for the multi-bolted connection type B is shown in Figure 4. Four element sizes are chosen to ensure that a dense mesh is obtained in the vicinity of the bolt holes and a less dense one at the composite plate free edges.

As the load is gradually applied to the specimen the lower gap elements close and the upper ones open. These gap elements are fictitious in that they offer no resistance to the relative motions of two adjacent continua, except to prevent those two regions from overlapping under the applied load. Therefore, the adjacent regions under the applied load are free to deform such that a portion of their boundaries may be in contact, but may not cross. This is achieved mainly by assigning to these gap elements, which are fully closed, a modulus sufficiently large to allow contact of the neighbouring boundaries while preventing overlapping of these boundaries. Maximum displacements in the direction of fibers, Uy ,

occurred at the top of the plate where the load is applied. Also, maximum displacement in the transverse direction occurred at the elements near the edge of the plate at the section of the most stressed bolt, which is the upper one. This verifies the test results which indicate that the inner most bolt is the most loaded bolt. The analytical load­displacement relationship is parallel to the measured

50'~--------------------------------------------------------------------------,

Max. Load = 214.12 kN w=8", e=4" Fibres @ 90 deg.

, .. -

o 2 3

···~-·---f·-·-·--?-···-·--····-·-·-··-·-I 'w ' I Joint B

( - Experimental ........... Finite Element J

456

Displacement (mm) 7 8 9

Figure 5 Experimental versus finite element results for load-displacement at 90° fiber orientation (pattern B)

10

345

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

(a)

K

Zw,

~~______________ -2y, Y,v,

(b)

~ yGap

J

~--------------------------- y

(c)

~--------------------------- y

Figure 2 Element types

represents two surfaces which may maintain or break physical contact and may slide relative to each other. The geometry and node locations are given in Figure 2(b). The element is defined by two nodes and the interface is assumed to be perpendicular to the I-J line as shown in Figure 2(b).

Since at the start position these gap elements are in open status, a three-dimensional spring element of very small stiffness is associated with each gap element. The spring element (stiff 14) has longitudinal or torsional capability in one-, two- or three-dimensional applica­tions. The geo,metry and node locations for this element are shown in Figure 2(e). The definition of all these types of elements is given in the ANSYS manual9

.

. As for the boundary conditions shown in Figure 3, the area of the steel bolts is held fixed due to the high rigidity of the bolt compared to the GFRP plate (nearly 16 times). The bottom and the left edges of the plate are free. The load was applied in terms of pressure per length (width of the specimen) on the top edge of the plate. These boundary conditions simulate exactly the con­ditions during experimental testing of the bolted con­nection using the MTS machine. By taking advantage of

344

I C.L.

4

PF

4

3 jC.L.

Boundary Conditions

: Applied load 2,3,5: Free 6 : Fixed 4 : X direction simply

supported

Bolt

Figure 3 Boundary conditions

the symmetry about the Y-axis, only half of the plate is analyzed. As a result, the model represents a rigid frictionless bolt and the contact surface between the bolt and the laminate is semicircular.

Due to the high percentage of the randomly distributed fiber plies, the laminate is analyzed as one transversely isotropic layer with the assumption that the material is elastic and nonlinear.

The stress components, obtained by the finite element analysis at the centroid of each element around the bolt hole, are used to predict the maximum allowable stress in the laminate through the use of the tensor polynomial failure criterion. For three-dimensional analysis, the Tsai-Wu polynomial used in the program is:

(~ + ;,)ux + (~ + ;,)uy + (~+ ~,)uz - ;J, 2 2 2 2 2 F

__ U_y _ _ _ u_z_ + _ux_y + _uy_'z + _ux_z + xyUxUy (14) YY' ZZ' S 2 Q 2 R 2 -Jr=X""x'=;':=Y=Y"'"

+ Fyzuyuz + Fxzuxuz = ¢ Jyy'ZZ' v'XX'ZZ'

where ¢ is the output of the Tsai-Wu failure criterion, X, X' are the tension and compression failure stresses in the x-direction, Y, y' are the tension and compression failure stresses in the y­direction, Z, Z' are the tension and compression failure stresses in the z-direction, Fxy is the X - Y coupling

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

tensile and compressive strengths, the strengths along the fibers are different from those in the transversal direction. Therefore, there are four uniaxial failure strengths X, X', Y and yl for a composite material, in addition to the shear strength which is also independent. This makes a total of five strength characteristics. The objective of the failure criterion is to select an envelope that will define the strength of an orthotropic ply under combined stresses, since all plies in a laminate are under combined stresses. The key issue in comparing the merits of Tsai-Wu failure criterion6 to other criteria is the interaction among the stress components, which are independent material properties in the Tsai-Wu criterion, while they are fixed (not independent) in Hill's criterion 12. In the maximum stress or the maximum strain criterion, six simultaneous equations are required and interactions are not admissible I I .

The basic assumption of the strength criterion is that there exists a failure surface in the stress space in the following scalar form:

f(O"d = FiO"i + FijO"iO"j = 1 (10)

where the contracted notation is used and i, j and k = 1-6. Fi and Fij are strength tensors of the second and fourth rank, respectively. The linear term (O"J takes into account internal stresses which describe the difference between positive and negative stress-induced failures. The quadratic term (O"i O"j) defines an ellipsoid in the stress space. Higher order terms (Fijk ), (O"iO"jO"k) are ignored to avoid open-ended failure surfaces. Certain stability conditions are incorporated into the strength tensors, where the magnitude of the interaction terms Fij are constrained by the following inequality:

(11 )

where i,j = 1, ... ,6. Geometrically this inequality insures that the failure

surface will intercept each stress axis and the shape of the surface will be ellipsoidal, and not open-ended like a hyperboloid. When expanding equation (11) in a matrix form and for a triclinic material in three-dimensional space, the F i , Fij components would be:

Fi=

Fij =

FI

F2

F3

F4

F5

F6

FI

also

FI2

F22

(symmetric)

FI3 FI4 FI5 FI6

F23 F24 F25 F26

F33 F34 F35 F36 (12)

F44 F45 F46

F55 F56

F66

If a material has some form of symmetry, some of the interaction terms will vanish. For specially orthotropic materials, the terms F4, F5, F6 will vanish. Also, the coupling between normal and shear strengths will vanish, i.e. FI6 = F45 = F56 = F46 =. zero. But the coupling between the normal strengths will remain.

For transversely isotropic material with the 2-3 plane as the isotropic plane, F2 = F3, FI2 = F13 , F22 = F33 , and F55 = F66 , also F44 = 2(F22 - F23)' Thus, for transversely isotropic material the Fi and Fij components are:

Fi=

FI

F2

F2

0

0

0

F1]

also

FI2 FI2

F22 F23

F22

(symmetric)

0 0 0

0 0 0

0 0 0

2(F22 - F23 ) 0 0

F66 0

( 13)

Therefore, the number of independent components reduces to 2 and 5 for Fi, Fij' respectively.

Connection modelling

The 8-node layered shell element was used to idealize the GFRP composite plate and the high strength steel bolts used for the bolted connection. This element (stiff 99) has six degrees of freedom at each node: translations in the nodal X, y, Z directions and rotation about the nodal X, Y, Z axes. The geometry, node locations and the co-ordinate system for this element are shown in Figure 2(a). The element is defined by 8 nodes, average or corner layer thickness, layer material direction angles and orthotropic material properties.

There are several possible ways to treat the bolt/hole contact problem in the finite element model. One is to assume a certain contact pressure distribution acting along the boundaries of loaded holes, for example a cosine distribution; another is to assume radial displacements equal to zero on the hole boundary, which yields a distributed contact reaction to the applied load. Performing a complete analysis is a third possibility, which obviously is the most accurate. In this investigation, contact stresses and stresses in the vicinity of the hole are calculated with account taken of the contact problem. The clearance between the hole and the high strength steel bolt was represented by a three­dimensional gap element. The contact element (stiff 55)

343

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

where u;, V;, w; are the global nodal displacements, Bxi ,

By;, Bz; are global nodal rotations, ti is the nodal thickness, ( is the local co-ordinate in the through thickness direction, and, at, aiv , at are the direction cosines of the u, V and w directions with respect to the nodal co-ordinate system at node i for the vector a.

In the isoparametric formulation the co-ordinates of a point within the element are obtained by applying the element shape functions to the nodal co-ordinates:

n

X(e) = 2:)N;} (e) {Xi}(e) i=l

n

y(e) = 2:)N;} (e) { y;}(e) (3) ;=1

n

Z(e) = 2)N;}(e){z;}(e) ;=1

where for a node (i) of element (e), {Ni}(e) are the matrix shape functions and they are the same shape functions use<;l in the element displacement field, (n) is the number of nodes per element, and {X;}(e), {y;}(e), {Z;}(e) are the nodal co-ordinates.

Stress-strain relationship

The generalized Hookes law relating stresses and strains for a linear elastic anisotropic and homogenous material (triclinic symmetry) is:

[0'] = [O][E], which on expansion yields

O"x D\l Dl2 DI6

f~

O"y D2l D22 D26

Ey

O"z D3l D32 D36

Ez (4)

Txy "txy

Tyz "tyz

Txz D6l D62 D66

"tzx

where { O"}, {E} are the stress and strain vectors, respectively, also:

[E] = [SHO'] (5)

The matrices [0] and [S] are the stiffness modulus and the compliance matrix, respectively.

However, a complete constitutive description of a general anisotropic solid necessitates the experimental evaluation of 21 elastic independent constants out of the 36 constantsd . If any material symmetry exists, the number of constants will reduce. Therefore, for a monoclinic material the 1-2 plane or z = 0 is the plane of symmetry and the number of independent constants reduces to 13 out of20 non-zero constants. As the level of material symmetry increases, the number of independent constants continues to reduce. If symmetry exists in three orthogonal planes, as is the case for orthotropic material, the number of independent constants becomes 9 out of 12 non-zero constants, provided that the planes of symmetry coincide with the reference co-ordinate system. But if the

342

axes of the reference co-ordinate system are not coincident with the symmetry planes, the number of non-zero constants will be the same as for the triclinic material. The next level of material symmetry is the transversely isotropic material, which has 5 independent constants out of 12 non·zero components.

Since the glass fiber reinforced plastic sheets used in this investigation are formed of alternative layers of unidirectional E-glass fiber rovings and randomly oriented E-glass continuous strand mat, the material is considered to be transversely isotropic. The stress-strain relationship is defined as:

[O'](e) = [O](e)[E](e) (6)

Therefore, within each element the relationship is:

[O'](e) = [0] (e) [B](e) [q](e) (7)

where for transversely isotropic material, the material stiffness matrix [0] is:

Dll Dl2 D\3 0

D21 D22 D23 0

D3l D32 D33 0 [0]=

0 0 0 (D22 - D23 )

2 0 0 0 0

0 0 0 0

where Dij are the stiffness coefficients:

DII = (1 - Vi3)AEl

D22 = D33 = (1 - V21 VI2)AE2

DJ2 = DI3 = V21 (1 + v23)AE2

D23 = (V23 + v21 v12)AE2

E2 D44 = (1 - V23 - 2VI2V21)A2

D55 = D66 = G12

0

0

0

0

D55

0

A= 1 (1 + v23)(1 - v23 - 2v21v12)

0

0

0

0

0

D66

(8)

(9)

where EI is the section modulus in the direction of the fibers, E2 is the section modulus in the transversal direction to the fibers, and Gl2 is the shear modulus.

Failure criteria

For the purpose of material characterization and design, a simple strength criterion for composites is essential. Strength covers many aspects associated with the failures of materials, such as fracture, fatigue and creep, under quasi-static or dynamic loading, exposed to corrosive environment and subjected to uniaxial or multi­axial stresses in two- or three-dimensional configurations. Failure of composites is further complicated by a multitude of independent and interacting mechanisms, which include fiber breakage, micro-buckling, delamina­tion and crack propagation. In addition to the different

Finite element analysis of bolted connections for PFRP composites: Nahla K. Hassan et al.

Table 1 Finite element specimens

Edge Fiber Specimen Width distance angle

. Joint pattern designation (in) (in) (degrees)

Single OS15 6 1.5 0

45S15 6 1.5 45

90S15 6 1.5 90

A OA7 8 1.5 o

OA9 8 4.0 o

45A3 8 4.0 45

90A6 8 4.0 90

B OB3 8 4.0 o

45B3 8 4.0 45

90B3 8 4.0 90

Thickness (I) = 0.5in, Pitch (p) = 3.25in.

Table 2 Material properties

Tensile Compressive Shear Thickness (mm)

Fiber angle (deg.)

Tensile modulus (GPa)

Elongation strength strength strength

12.7 o 90

12.7 10.5

Shear modulus = 4.3 GPa; Major Poisson's ratio = 0.29.

(%)

1.6 1.4

and the strength properties of the specimens studied are shown in Table 2.

Due to the high percentage of the randomly distributed fiber plies, the laminate is analyzed as one transversly isotropic layer with the assumption that the material is elastic and nonlinear.

Displacement model

In general, the distribution of the displacement over the domain of the element and the formulation of the displacement model is achieved by using either a simple polynomial model in terms of generalized co-ordinates or an interpolation model. It is possible to avoid computational work by selecting an interpolation function as the basis for the displacement model. This interpolation function, also known as a shape function, has a unit value at one nodal point and zero value at all other nodal points. Thus, with inter­polation models the analysis will include directly the nodal displacements rather than the generalized displacements. In this study the three-dimensional shell

(MPa) (MPa) (MPa)

166 175 27.8 110 145 24.0

element's shape functions for the corner nodes and the mid-side nodes, respectively, are:

Ni = * (1 + ~~J(l + 7777i)(~~i + 7777i - 1)

Ni =!(1-e)(l+7777i) at~i=O

Ni = !(1-772)(1 + ~~J at 77i = 0

(1)

where ~, 77, are the local co-ordinates in the center of the element.

The element displacement field {u} e can be expressed in terms of nodal translations and rotations and the shape function {NJ e as:

(2)

341

Finite element analysis of bolted connections for PFRP composites: Nah/a K. Hassan et al.

One of the problems in multi-bolt joint design is that the load is not equally distributed around the bolts, especially in joints having more than two rows of bolts. Thus there is an interaction of stress fields around the bolts which is not fully understood or determined. In this paper, a three-dimensional finite element analysis using the ANSYS program9

, was used to perform stress analysis of single and multi­bolted double shear lap connections of glass fiber reinforced plastic, taking into consideration the bolt­hole contact problem. The Tsai-Wu tensor poly­nomial has been applied as the failure criterion to predict the ultimate failure strength of the composite bolted connection. The predicted load distribution among the bolts and the predicted strains using the finite element program were compared with the experimental measured values and good agreement was achieved.

OVERVIEW ON THE EXPERIMENTAL OBSERVATIONS

The behaviour of bolted joints is highly dependent on the geometric dimensions of the connection, including the edge distance, width and pitch between the bolts, in addition to the number of bolts and the bolt pattern. The experimental investigation conducted at the University of Manitoba to study the behaviour of multi-bolted connections of five different configurations, including test results, modes of failure, and load distribution among the bolts, are included in another paper. In this investigation, two basic failure modes consisting of net tension and cleavage failure were observed for the different geometric dimensions considered in the experimental program 10. Test results indicated that the net tension failure was characterized by sudden crack propagation and hence failure was catastrophic in nature. The cleavage failure mode on the other hand, had a much more ductile behaviour. All the connections tested showed signs of bearing mode damage in the vicinity of the bolt hole after failure, where damage occurred in the laminate area adjacent to the loaded half of the hole. Some of the important observations and conclusions derived from the experimental program are summarized as follows:

(1) The ultimate load capacity of the five different configuratidns of the multi-bolted connections, shown in Figure 1, increased by increasing the geometric dimensions of the member. However, it was found that increasing the edge distance (e) to hole diameter (d) ratio (e I d) beyond 5 or the side distance (s) to pitch (p) ratio (sip) beyond 1.2 had an insignificant effect on the ultimate load capacity of the connection.

(2) The load-strain relationship for all types of con­nection is linear up to failure. For connections with one row of bolts, the load is equally shared among

1 - ----,.---- -.

1 , ----$+----

1

tJointA

1 1 - -...,.----or- -.

1 1 1 , P'P'

~

3 iii' f~-$+Ei>t-If

1 12 I 11"

JointD

! P ! ,-, 1

JointB

- --+---·1- .. . I, I, f-t-~-~I- -

--~---$r -~2

JointE

Figure 1 Connection configuration

- ----1------.

Joint C

the bolts. For connections with more than one row of bolts, the bearing forces of the fasteners are not evenly shared as measured by the gauge readings.

(3) The increase in the ultimate capacity of the con­nection is not directly proportional to the increase of the number of bolts. The effect was more significant when increasing the number of bolts in a column than when increasing the number of bolts in a row.

(4) Variation of the bolt pattern has a minor effect on the gross efficiency of the connection, when com­paring connections with the bolt in a row to those with the bolts in a column (same number of bolts in each connection).

FINITE ELEMENT ANALYSIS

Problem definition

The single bolted connection at three fiber orienta­tions was investigated analytically using the ANSYS program. In addition, one specimen of each type of the two different joint configurations at 0°, 90° and 45° fiber orientation, previously tested experimentally, were studied as given in Table 1.

The fiber reinforced composite material used in this investigation is Extren flat sheet series 500, a pultruded glass fiber sheet produced by Morrison Fiber Glass Company (MMFG). The composite material is ortho­tropic, consisting of symmetrically stacked alternating layers of identically oriented unidirectional E-glass rovings and randomly oriented E-glass continuous strand mat. The mat provides multidirectional strength properties, whereas the continuous rovings provide strength in the longitudinal (pultruded) direction. The matrix consists of isophthalic polyester plastic, which is a general duty resin that is resistant to most acidic attacks, and the fiber content is approximately 40%. The material

1359-8368(95)00046-1

Composites: Part B 27B (1996) 339-349 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved

1359-8568/96/$15.00

Finite element analysis of bolted connections for PFRP composites

Nahla K. Hassan*, Mohamed A. Mohamedient and Sami H. Rizkallat * Structural Engineering Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt t Civil Engineering Department, Faculty of Engineering, Suez Canal University, Port-Said, Egypt :\: Department of Civil Engineering, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada (Received August 1995; accepted October 1995)

Structural applications of composite laminates are increasing with the drive for high strength, lightweight component design. These applications usually require joining composites either to composites or metals. Most commonly, joints are formed using mechanical fasteners. Therefore suitable prediction methods must be developed to determine failure strength and failure modes of these bolted connections.

In this paper, three-dimensional finite element analysis is conducted on single and multi-bolted connections to determine the failure process, ultimate load and load distribution among the fasteners, taking into consideration bolt-hole contact problem in the finite element model. The strength analysis is based on Tsai-Wu tensor polynomial failure criterion which is applied to the laminate as a whole.

(Keywords: composite materials; composite laminates; bolted connections; finite element; tensor polynomial failure criterion)

INTRODUCTION

With the advent of composite materials and their wide use in highly stressed lightweight constructions, it has become necessary to develop a deep insight into the failure and strength of composite bolted joints. The strength of such joints depends on the number and spacing of bolts, edge distance, bolt clearance, load distribution among the bolts and material properties. Current design procedures for mechanical joints of orthotropic materials tend to be empirically based and exist for only simple cases. Insufficient knowledge of the stress distribution in the neighbourhood of loaded holes in anisotropic materials and their interaction with the material properties have inhibited development of adequate design theory. For this reason, many finite element procedures have been proposed for analyzing composite bolted connections. Agarwal l used finite element technique and the average stress criterion over a characteristic distance d (from the hole boundary) as the stress level to predict the failure strength of the laminate. The parameter d is considered as a material property, Garbo and Ogonowski 2 used the characteristic distance concept of Whitney and Nuismer3 to determine the strength of composite laminates with unloaded fastener holes. According to this hypothesis, they chose

a characteristic distance d away from the hole and used the stress level at this distance to compute the failure loads of different laminates by the Tsai-Hill criterion4

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The average stress criterion considers only one stress component for predicting the strength of the laminate. In fact, at almost all the points around the hole boundary, a biaxial state of stress exists. Further, with the change in ply orientation of the laminate, the points of stress concentration also change. Tensor polynomial failure criterion has been used, because it does not require additional constant d, and all the stress components are accounted for in this criterion. Sonis used Tsai-Wu6

tensor polynomial failure criterion to predict the ultimate failure strength of single bolted joint for 00 ± 45° lay-ups and 90° ± 45°. York et at.7 predicted the net tensile strength of composite bolted joints using the Sap-V finite element computer program using a two dimension FEA employing quadrati cal elements with orthotropic material properties. Wang and Hann8 in 1988 used a general purpose finite element program NF AP, where the plates were modeled by the use of eight noded plane elements with anisotropic material proper­ties. As for the fasteners, they proposed two-dimensional pure shear elements and undergoing pure shear strain along the distance between the middle surfaces of the two plates.

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