0045-7949(94)00447-l

Compu,m & Srrmrurrs Vol. 55. No. 2, pp. X7-363. 1995 CopyrIght 3; 1995 Elsev~er Saence Ltd

Printed in Great Bntam All rights reserved 0045-794919s $9.50 + 0.00

ANALYSIS OF A DOUBLE SHEAR LAP JOINT WITH

INTERFERENCE FIT PIN

N. Sundarraj,? B. Dattagurut and T. S. Ramamurthyz

Post, Bangalore- 017, India tAeronautica1 Development Agency, P.B. 1718, Vimanapura $Department of Aerospace Engineering, Indian Institute of Science, Bangalore- 012. India

(Received 4 February 1992)

Abstract-This paper presents the study of stress distribution in a double-shear lap joint with interference fit pin subjected to inplane plate loads using FELJNT, a finite element software developed for this special purpose. FELJNT accounts for the effects of clamping and flexure of the bolt without having to adopt a laborious three-dimensional analysis. It carries out stress analysis in an axisymmetric region around the pin under non-axisymmetric loading. It uses an iterative method of contact stress analysis and a dummy element concept with frontal solver for finite element solution. This paper discusses the load for initiation of separation, through-the-thickness stress distribution, and qualitatively the beneficial effects of inter- ference on fatigue life of the joint. The paper concludes with comparison of FELJNT results with those of JACKAL. a three-dimensional finite element analysis software for joints.

INTRODUCTION

Bolted joints are common in mechanical and aerospace structures. The most crucial in aircraft structures are the wing-root and fin-root fittings. A typical joint consists of two or more plates with circular holes joined by multiple fasteners. In such joints the interference fit bolt gives improved fatigue performance. Such benefits can be qualitatively predicted by a two-dimensional (2-D) analysis of each plate with an interference fit pin. Such 2-D analysis cannot account for the effects of bolt clamp- up force and bolt flexure during deformation and they can only be catered for in a three-dimensional (3-D) analysis.

Let us consider a double-shear single-bolted joint consisting of three plates connected by a bolt and a nut. Using 2-D analysis the stress distribution in each plate and the partial contact behaviour at pin-plate interface have been extensively studied in the past [l-6]. These analyses provide a rigorous contact stress analysis to deal with the non-linear problem arising out of partial contact. Harris et al. [I ] used an iterative finite element method and solved the problem of interference fit pin in a rectangular plate subjected to pin load. Eshwar et a/. [2] used an inverse method to study load contact behaviour of a rigid pin in an infinite isotropic plate. Ghosh ef al. [3], and Naik and Crews [4] solved the same problem as in Ref. [2] with an inverse method for elastic pin inclusion. Mangalgiri et al. [5] studied interference, push and clearance fit joints in orthotropic plates under bearing and by-pass loading. Using the incremental iterative finite element formulation, Brombolich [6] studied the effect of friction in interference fit pin joints. The non-linear

contact stress analysis was carried out in these con- tributions by either iterative or inverse method. In the former the contact/separation regions are iteratively determined for a given load level, whereas in the latter, load levels required are determined for vari- ous extents of contact/separation. Inverse method could be economical, but is restricted in its appli- cation. The iterative method is general, but could be computationally expensive. It requires special atten- tion and development of algorithms to decrease the computational effort. Sundarraj et al. [7] developed a contact stress algorithm using a frontal solver and a dummy element concept for this purpose. This method was shown to be effective for 3-D analysis of bolted joints. A software named JACKAL was devel- oped using this technique.

In spite of the developments mentioned above, 3-D finite element analysis is laborious and generally un- economical. Three-dimensional effects were studied using axisymmetric analysis by Thompson et a/. [9] and Novikova [IO]. Thompson et a/. [9] analysed the effect of clamp-up pressure on circular plates clamped between two rigid circular punches. The bolt-clamping effect was introduced by assuming a uniform indenta- tion of the plates at the bolt-head and the nut regions. Novikova [IO] studied bending stresses on the bolt for an in-plane plate load and carried out analytical studies on the bending of the bolt. A three-dimensional photoelastic stress freezing technique was used to determine bolt stresses and these were compared with analytical results.

Combining some of the concepts in literature, it is possible to study 3-D effects in a double-shear single- bolted joint by analysing the axisymmetric region around the bolt hole subjected to a non-symmetric

357

358 N. Sundarraj et al

plate load at its boundary. FELJNT, a finite element software for contact stress analysis of interference fit joints has been developed for this purpose [l I]. It models the reactive load at the boundary of the axisymmetric region around the hole as equivalent Kelvin’s harmonic stress distribution, and the clamp- up pressure as a parabolic distribution at bolt-nut interface [ 121.

This paper presents the results of the study of load transfer in a smooth interference-fit, double-shear lap joint with and without clamp-up pressure, using FELJNT.

DEFINITION OF PROBLEM

Figure 1 shows a double-shear joint consisting of three large plates A, B and C of thicknesses 7’,, TZ and r,, respectively, connected together with an oversized bolt of diameter 2n (I + i, ) in a hole of diameter 2a and a nut of outer diameter 26, where ,I is the proportional interference (=jdiameter of the bolt-diameter of the hole}/diameter of the hole).

A clamp-up load P is introduced into the bolt by applying a specific torque on the nut. The clamp-up load would cause compressive contact Stresses over a contact radius along the interface between the plates, and sepuration over the remaining region. Besides, the joint transfers in-plane loads from one plate to another. A tensile force F on the middle plate is reacted to by forces of magnitude F/2 on each of the outer plates. The plate load is reacted as a distribu- tion of contact pressure at the plateebolt interface. The integrated sum of interfacial contact pressure distribution should be equal to F/2 in outer plates and F in the middle plate. When the plate load increases from 0 to F, the bolt which is initially

Fig. I. A typical double shear bolted joint Fig. 2. Finite element mesh pattern.

straight will bend due to bearing pressure between the bolt shank and the plates. This configuration of the bolted joint is to be analysed for stresses under the combined effect of the clamp-up load P and the pin-bearing load F.

The crucial aspects of the present analysis are:

(i) Since the plates are assumed to be large, the clamp-up load P and its effect on the joint (in the absence of pin-bearing load F) will be axisymmetric. The plates will be in contact only over a circular region of radius c, and will lose contact over the remaining region. This contact region is unlikely to be significantly affected by the pin-bearing load F. So while considering the combined effect of P and F, it is assumed that the contact region remains circular.

(ii) The analysis will be carried out over an axi- symmetric part of the joint whose outer radius is 4-6 times larger than the radius c of the expected annular contact region between the plates.

(iii) The bolt is considered to be an interference fit, so that the contact is maintained all along the bolt- plate interface. If the contact is lost over a partial region, the axisymmetry of the problem will be lost. So, the current analysis would be applicable only for loads which maintain full contact along the bolt- plate interface (i.e. up to initiation of separation).

Since the plates are assumed to be large, the details of load transfer at bolt-plate interface would not be affected by any statically equivalent loading at the outer axisymmetric boundary. Solution to the problem of point-loaded infinite plate is available in literature [13]. The normal and shear stresses from this solution are introduced as the load applied on the outer boundary. Such a loading is non-axisymmetric and will be dealt with by a harmonic analysis.

Smooth (zero friction) surface conditions are assumed so that the axisymmetric field is not disturbed. This implies that the load transfer is prim- arily along the load-bearing surfaces between the bolt and the holes. Joints where the effect of friction is significant would need a 3-D analysis.

FINITE ELEMENT MODEL

The outer diameter (2s) of the axisymmetric part of the joint modelled is 20 times the diameter of the hole. The axisymmetric domain of the plates and

Double shear lap joint with interference fit pin 359

the bolt is modelled by eight-noded axisymmetric ring elements. A typical mesh (zoomed) near the bolt hole is shown in Fig. 2. Pairs of nodes are considered at all the interfaces in the joint to perform contact iterations. A typical finite element mesh selected for analysis after a systematic convergence study has about 1200 degrees of freedom. The mesh, boundary conditions, and the consistent load vector for the specified loading are generated by using FELJNT. The software also generates a fictitious dummy element connecting all the pairs of nodes at various interfaces required for contact stress analysis and nodes where external load is applied.

METHOD OF SOLUTION

As mentioned earlier, the domain of analysis is axisymmetric, and the plate load non-axisymmetric. The solution can, therefore, be in two parts, viz., axisymmetric for clamping and interference loads, and non-symmetric for plate load. The solution can be obtained by iterative contact stress analysis pro- vided by the software using modified frontal solver reported by the authors [7]. In this method the frontal solver has been used to eliminate the degrees of freedom not required for iterations, and a dummy element concept was used to facilitate the elimination process. The present software uses similar approach for the axisymmetric region.

All the interfaces in the joint except the bolt-shank plate boundary are examined for contact/separation region, so as to get the axisymmetric solution. The non-axisymmetric load and the displacement solution are expanded into their Fourier components. For each component of the series, a corresponding solution can be obtained using FELJNT.

The displacement functions are expressed in the form of a harmonic series. Each term in the displace- ment function corresponds to a harmonic part of the loading.

For the present problem with one axis of sym- metry, the displacements at any point in the domain, may be expressed in the series form as

[U] = [Uo] + C U,cosjO (1) ,= 1

[V]=[VO]+ i V,sinjQ (2) ,=I

[W]=[W,]+ i WicosjQ. (3) ,=I

The constant terms U,, V,, and B’, in the series are the radial, tangential and axial displacement components of the axisymmetric part of the solution, and U,, V, and W, are the corresponding amplitudes ofjth harmonic part of the solution.

The elements in the (T, z) plane could be trans- formed into the natural coordinate system (5, q).

The shape functions N,s for the displacements are expressed in terms of 5, rl and harmonic functions of 0. For an axisymmetric domain the displacement distribution corresponding to the jth harmonic solution can be written as

b’l= NA5, rl)bl’cos.N (4)

[v’] = N,(<, q)[u]‘sin j@ (5)

[w'l = N,(4, rl)[wl” cosj& (6)

where [u]e, [a]’ and [wF correspond to the nodal values of the displacement corresponding to the jth harmonic of the loading.

The loading for the jth harmonic is given by

[R] = R, cos j0 (7)

[T] = T, sinjO (8)

[Z] = z, cosj8, (9)

where [RI, [T] and [Z] are the consistent load vectors in radial, tangential, and axial directions and R,, T, and Z, are the amplitudes of thejth harmonic forces.

The pin-bearing load per unit thickness P, is reacted at the edge of outer axisymmetric region by the or, alo stress system derived from the classical Kelvin’s solution of a point-loaded infinite plate [13]. These stresses are given by

PY(3 + v) (T I = --------COSB

4xr

Px(l - VI sine

fJ,u = 4nv 9

(10)

(11)

where v is Poisson’s ratio. Using the above stresses. the equivalent nodal loads on the three plates are generated in the form given in eqns (7))(9). Since the boundary loads are first harmonic components, the non-axisymmetric part of the solution corresponds to the first harmonic and there will be only two terms in the final solution.

In the present problem the displacements at any 0 are given by

[1(] = ug + u, cos 0 (12)

[o] = c0 + c, sin 0 (13)

[W] = M’” + U’, cos 0, (14)

where z+,, ~1~ and w,, are the axisymmetric components and the rest are the non-axisymmetric components of the displacement solution. At any location (r, 0, r) in the domain of analysis, the solution is obtained as the sum of axisymmetric and harmonic terms.

Boundary conditions are applied to take care of the rigid-body displacements and rotations in the prob-

360 N. Sundarraj ~‘f trl

lem. For this purpose node P (as shown in Fig. 2) at the free end of the bolt head is restrained for r- and -_-displacements, and node Q at the free end of the bolt shank is restrained for r-displacements. Contact boundary conditions arc imposed on the appropriate displacements in the axisymmetric and harmonic parts of the solutions.

INTERFERENCE FIT JOIYTS

Interference in joints has beneficial effects. The introduction of interference fit in joints causes initial compressive stresses in the joint. As the pin load is progressively applied, the pin-hole interface main- tains full contact only up to a certain load level. As the load is increased further. the interface exhibits a loss of contact. i.e. it undergoes separation. Till the initiation of separation. the stresses in the plate vary linearly with load level. Beyond this point. the extent of separation varies with load and consequently the problem is non-linear.

When the joint is subjected to cyclic loading. the fatigue life of the joint is dependent on the cyclic variation of local stresses around the boundary of the hole. Such a variation is less for an interference fit compared to neat or clearance fit pin joints. This explains the beneficial effects of interference on fatigue life of joints. Howcvcr, these beneficial effects are considerably reduced when the interference fit operates in post-separation range of loading. So, in order to gain significant]) from the use of intcr- ference, one should look for (a) high load levels for initiation of separation (P, ),, and (b) low values 01 variations in hoop stress (u(,) with applied load P,. i.c. (?w,,,‘2P,) up to the separation load Icvcl. So. the numerical results will bc discussing these two parametric values. viz. (P, ), . and (c?cr,, ;a P, ).

NLMERICAI. EX.4MPI.ES

The numerical example considered is a three-plate lap joint (dimensions r, = 2T, = 2T, = 5 mm, 2tr = IO mm. 2h = 20 mm and 2d = 200 mm) with an inter- fercncc bolt of i = 0.005 between the bolt and the hole. The joint is subjected to a load Fat the middle, and reacted by F:2 at outer plates. The clamp-up load

Fig. 3. Bearmg stress dstrlhution-- finger tight joint.

P = 0 kgf in case (i), finger tight joint, and P = 750 kgf in cast (ii), clamped joint. In the first example an aluminium bolt and aluminium plates are con- sidered so that E,,/E, = 1.0. These problems are investigated essentially to study the effects of bolt clamping and flexure, on separation load, and on through-the-thickness stress variation in the joint. Also, the effect of E,/E, on through-the-thickness stress distribution is studied. In order to demonstrate the computational efficiency of FELJNT, a sym- metric stud joint problem is solved by both software FELJNT and JACKAL.

Both finger tight and clamped joint stress distri- bution was solved for plate load F = 0 kgf and F = 200 kgf (400 kgf cm-‘). When the plate load is zero, the radial stresses are compressive around the bolt- hole interface due to the interference. When the plate load is non-zero, the bearing stress on one side of the hole increases and that on the other side decreases. This causes through-the-thickness variation in the plate bearing stress. At a particular plate load, the stress at a point along the hole surface becomes zero and this corresponds to the separation-initiation load F,,

Fingrr light joO7t. The bearing stress distribution for the two load cases of F = 0 kgf and F = 200 kgf for finger tight joint is shown in Fig. 3. The separation-initiation load for the finger tight joint is obtained by extrapolating the stresses from the two plate load cases of 0 and 200 kgf. The bearing stress distributions at the bolt&hole interface for the above plate loads are identified as curves I and 2 in Fig. 3. It is seen that the through-the-thickness stress distri- bution in the ccntre plate is nearly uniform for the case of pure interference. When the plates are loaded, there are large variations of through-the-thickness stresses in all the plates caused significantly by the bending of the bolt. There is a marginal asymmetry about the centre of the middle plate, since on one side the plates arc ciamped by a bolt and the other side through the nut, and the load transfer is different at the two ends.

The interfaces between top and middle plate and bctwccn middle and bottom plate are referred to as plane I and plane 2 (Fig. 2). The interfaces between the bolt head and the top plate and between the nut and the bottom plate arc planes 3 and 4 (Fig. 2). The cffcct of bending of the bolt would be large reduction in compressive stresses near planes I and 2. By linearly extrapolating the results between plate load F = 0 kgf and F = 200 kgf. it is found that the separation is initiated at plane 2 in plate B at 684 kgf (I 368 kgf cm ’ ). For this load level the bearing stress distribution is identified by label 3 in Fig. 3.

C’/rrnl@ joinr. Clamping of the plates at the bolt head and nut introduces further changes in the

Double shear lap joint with interference fit pin 361

Fig. 4. Bearing stress distribution-clamped joint.

Fig. 5. Deformation of the joint

through-the-thickness stress distribution. Even in the absence of plate load, considerable amounts of bend- ing loads are introduced into the plate by clamping. Following the extrapolation procedure for the case of finger tight joint, the load for onset of separation in this joint is obtained as 41 I kgf (822 kgf cm-‘) at plane 2 in plate C. The bearing stress distributions for zero plate load and for separation load (F = 41 I kgf) for the clamped joint are shown as curves 1 and 2,

2.5 r

Table I. Comparison of load for onset of separation ti. = 0.005)

Finger tight Clamped Pm joint joint joint (3-D)

(2-D) (2-D) (P = 750)

Separation load /c, 2149 1368 822 Gradient a~,, :aF 0.1497 0.3194 0.1825

respectively, in Fig. 4. A typical deformation pattern (zoomed view) of the clamped joint is shown in Fig. 5. The gaps and overlaps seen in this figure between various components arc not real, they appear only due to magnification and rigid body movement of the bolt- plate-, and nut-deformations.

The load for initiation of separation obtained from 2-D study [3] is 1374.5 kgf (2749 kgfcm ‘). This is as expected. since in 2-D studies, all the points at the bolt-hole boundary along the thickness are assumed to separate simultaneously, whereas in the 3-D study this point corresponds to initiation of separation at any one point in plane I or 2.

The variation of maximum hoop stress and bearing stress with load is shown in Fig. 6. In this figure the stresses are normalized with 2-D value of interference radial pressure o,, and the plate load F is normalized with the value of EuiTz. The separation initiation load per unit thickness./;, and the hoop stress gradient i3o,,/ClF for the cases of 2-D pin joint, finger tight joint and clamped joint arc compared in Table I. It is observed that in the finger tight joint, the gradient of &r,,/aF is more than twice the value predicted by the 2-D solution.

In the presence of clamp-up load, the separation is initiated at a much lower load, so the change of linearity in stress distribution occurs at much lower load level than the 2-D predicted value. But the clamp-up load. though causing reduction in load for

Smooth interference fit joint

- Bearing stress

- - Hoop stress

-*- Initiation of separation

(3 5, = -!.!!!E

%

Fig. 6. Variation of stress concentration factor wrth plate load f

362 N. Sundarraj et crl

Table 2. Load for onset of separation for different bolt moduli (d = 0.005, P = 375)

E,iE,,

f;, kgfcm-’

t/3 I 3

246 1085 229 I

separation, decreases the gradient compared to finger tight joint. So the gradient &r,,/?3F is much smaller than that of the finger tight joint, and is closer to the value predicted by 2-D analysis.

&kt of bolt elusticity

The next aspect dealt with is the effect of bolt elasticity on the separation load and the through-the- thickness stress distribution. The loads required for initiation of separation for various values of E,/E, are shown in Table 2. Stiffer bolts have less bending and so the initiation of separation occurs at higher load. Since the stress levels decrease with increase in rigidity, the gradient aa,,/aF also decreases with increasing rigidity of the bolt, resulting in beneficial effects. When low modulus bolts are used, the operat- ing loads should be kept below the separation load level so as to enhance the fatigue life of the joints.

The configuration considered for comparison with 3-D analysis consists of a symmetric, three-plate lap joint connected by a stud and two nuts. A propor- tional interference i = 0.005, and a clamp-up load P = 750 kgf were applied on the joint. The problem was studied with respect to the cases with F = 0 kgf and F = 200 kgf. using FELJNT. This is identified as analysis (i). Through-the-thickness stress distributions were obtained. The separation load for this problem is obtained by linear extrapolation of stresses as 342.5 kgf.

The stud joint has symmetry about both z-plane and _tx-plane, and this symmetry reduces data hand- ling for 3-D analysis. The stud joint problem is solved using JACKAL. This is identified as analysis (ii). The separation loads in the two cases are compared in Table 3. The results compare within 1%. Though the loads compare very well, there are significant differences in the through-the-thickness variation of stresses. However, in most cases, the difference between maximum and minimum stresses is small. This data is not presented in this paper.

The numerical values of the differences between the distributions in the two analyses are quite crucial. If analysis (i) can predict nearly as well as 3-D analysis, then it is enough to use analysis (i) because it is far more economical than the 3-D analysis. Three-

dimensional analysis yields lower bearing stresses and higher hoop stresses than analysis (i).

Table 3. Comparison of load for onset of separation F,, in a stud joint-i. = 0.005, P = 750

Type of analysis F,, ( 15, ) Analysis (i) 342.5 kgf (685 kgfcm-‘) Analysis (ii) 346.0 kgf (692 kgf cm ’ )

x Hoa; ,,res, i&l, EwEP=I.O, T=, 0

%.=(I 005.P=7SO,F=3*6 EP=.7E+W. v=o.,

Fig. 7. Radial and hoop stress variation around the hole (plane 3).

The polar variations of the bearing and the hoop stresses from the two solutions are compared at different planes (plane I, middle plate; plane 3, outer plate) in Figs 7 and 8. The difference between these two solutions is higher in plane 1 of middle plate where separation is initiated. In plane 3, the difference between the two solutions is smaller. The maximum difference is about 17%. The maximum and minimum percentages of deviation in bearing and hoop stress

Fig. 8. Radial and hoop stress variation around the hole (plane I).

Double shear lap joint with interference fit pin 363

Table 4. The maximum and minimum deviation in stress CONCLUSIONS distributions obtained by analysis (i) and analysis (ii)

O(,l - U(U) % Deviation = ___ x 100

~hmrx(rr,

Cb Different planes max and min

Plane It 17.1 0.1 Plane 31 13.7 0.0

tMiddle plate; Souter plate.

Ch max and min

8.4 4.3 1.7 0.1

distributions at different planes of the joint are shown in Table 4.

A software tool FELJNT is developed and its suit- ability is demonstrated with respect to the analysis of double-shear interference-fit finger tight and clamped bolted joints. The software is based on analysis of an axisymmetric region around the bolt under non- axisymmetric loading. Three-dimensional effects due to bolt clamp-up pressure and bolt flexure and their influence on the through-the-thickness stress distri- bution in the joint are evaluated. The relative effort

required for this analysis compared to 3-D analysis is

brought out clearly.

The difference between the two analyses could be attributed to the loading system. In analysis (i), the load transfer in the joint is effected around the outer circular boundary. In 3-D analysis, the load transfer is different because the plates are considered rectangular. In axisymmetric analysis, the loading equivalent to Kelvin’s solution is introduced as a uniform stress distribution across the thickness of the plates. In 3-D analysis, the bending effects due to load transfer in lap joint make the distribution non- uniform across the thickness. The above differences could account for the deviation of 17% in the stresses, which can be considered very good for all practical purposes, particularly in view of the computational economy possible with analysis (i).

REFERENCES

1.

2.

3.

4.

5.

Computational economy

It is well known that the 3-D finite element analysis will be expensive. The relative computational efforts and the resources required to get the solutions from analyses (i) and (ii) are presented in Table 5.

An important observation from Table 5 is that 3-D analysis needs 254 times the CPU time and seven times the computer memory required by analysis (i). In addition, FELJNT, the software employed in analysis (i), is also portable to PC-based design tasks. However, it is to be noted that the application of FELJNT is limited to loads up to initiation of separation. Beyond the separation load, since the contact between bolt and the hole becomes partial, the joint can be analysed only through 3-D analysis.

6.

7.

8.

Table 5. Comparison of computational effort

Analysis (i) Analysis (ii) Ratio

Number of degrees of freedom 1218 11,856 9

Memory in k words 100 700 7 CPU time in seconds 45 I 1,440 254

*Analysis (i) and analysis (ii) were carried out in IBM 3090 computer.

9.

IO.

II.

12.

13.

H. G. Harris, 1. U. Ojalvo and R. E. Hooson, Stress and deflection analysis of mechanically fastened joints. AFFDL-TR-70-49 (1970). V. A. Eshwar, 8. Dattaguru and A. K. Rao, Partial loss of contact in interference fit joints. Aeron. J. R. Aerotz. Sot. Paper no. 643, pp. 233-237 (1979). S. P. Ghosh, B. Dattaguru and A. K. Rao, Load transfer from a smooth elastic pin to a large sheet. AIAA J. 19, 619-625 (1981). R. A. Naik and J. H. Crews, Stress analysis for a clearance fit bolt under bearing loads. Proc. 2tilh Structures, Structurul D_vnumics and Material Co& Orlando, FL, Part I, pp. 522-527 (1985). P. D. Mangalgiri, T. S. Ramamurthy, B. Dattaguru and A. K. Rao, Elastic analysis of pin joints in plates under combined pin and plate loads. Int. J. Mech. Sri. 29, 577-585 (1987). L. J. Brombolich, Elastic-plastic analysis of stresses near fastener holes. AIAA Paper no. 73-252, llth Aerospace Meeting, Washington, DC (1973). N. Sundarraj, T. Krishnamurthy, B. Dattaguru and T. S. Ramamurthy, An efficient algorithm using frontal solution for contact stress problems. Proc. In/. Con/: on FEA4 in Engineering, Melbourne, Australia, pp. I9 21 (1987). N. Sundarraj, A software tool for the analysis of bolted joints. Proc. National Seminur on Computer Applications in Defence (DEFCAP-88) Bangalore (1988). J. C. Thompson, Y. Sze, D. G. Strewel and J. C. Jofriet. The interface boundary conditions for bolted flanged connections. Trans. ASME, J. Prcwure Vessel Technol. 98, 277-282 (1976). S. I. Novikova, Strength of bolts in double shear allow- ing for rigidity of end-fixing. Rus.v. Engtzg J. LI, 7-8 (1972). N. Sundarraj, T. S. Ramamuthy and B. Dattaguru. FELJNT-a software for analysis of interference fit pin joints. Proc. Australirrn Aerospace Congr., Melbourne, Australia, pp. 637-646 (1991). N. Sundarraj, T. S. Ramamurthy and B. Dattaguru. A simplified model of load transfer in bolted joints. J. Aero. Sot. India 43, 29-34 (1991). S. P. Timoshenko and J. N. Goodier, Tlleorj, o/ Elasticity. McGraw-Hill, New York (1980).