A Numerical Approach on Load Sharing Analysis and Optimization of Bolted Joint Efficiency

Academic Open Internet Journal

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Volume 15, 2005

A Numerical Approach on Load Sharing Analysis and Optimization of Bolted Joint Efficiency

G.S.Shivashankar¹, S.Vijayarangan²

Research Scholar and Corresponding author¹,Principal²

Mechanical Engineering Department, PSG College of Technology, Coimbatore-4

Phone: 0422-2572177, 2572477, 2578455 Fax:91-422-2573833

E-Mail: gsshivashankar@rediffmail.com

ABSTRACT: The major design issue remains in progressing from the “coupon” strengths measured in these tests to bolted joints in different applications. A numerical method is used for the approximate determination of a multiple “bolted” joints loaded in tension and C code is developed for the same. Comparing with experimental results show that the method can be used as a first approximation for design purposes. In this failure analysis shearing effect, bending effect and bearing effect were considered. Effects of width/diameter ratio (w/d) and edge/diameter ratio (e/d) on failure load were estimated numerically and experimentally on A and C type joints of glass-fiber/epoxy composite laminates. The different stages of an analysis from plain hole to multi-row bolted joints consisting of

Distribution of load in the bolts and plate for a multiple bolt array numerically.
Prediction of stress concentration in pin-loaded hole.
Prediction of failure load for joints.

Finally the bolted joints can be designed based on the procedure developed. And also the failure strength of bolted connections of glass fiber woven mat/epoxy composite laminates with different staking sequence containing a circular hole was investigated numerically. Analysis has been done for different values of edge - to- hole diameter ratio (e/d ratio). Stress concentration is estimated in drilled laminates using Hart-Smith criteria of stress concentration. The influence of stress concentration on the efficiency of the joint was estimated numerically. Optimal value of e/d ratio is suggested for maximum efficiency.

Key words: Bolted joints; shearing effect; bending effect; bearing effect; numerically; efficiency.

1 INTRODUCTION

A laminate is formed by stacking on top of each other layers of reinforcement oriented in different directions. Composite materials, if properly used, offer many advantages over metals. Examples of such advantages are: high strength and high stiffness-to-weight ratio, good fatigue strength, corrosion resistance and low thermal expansion. It is more economical to build up complex shapes by joining simple geometrical shapes. Joining them is where problems tend to arise, which is one main problem concerning composite materials that plagues manufacturers. Thus, joining and attachment have been recognized as enabling technologies for successful utilization of composite components in various aerospace and ground based applications. Mechanical joints require that bolt or rivet holes are drilled into the composite that reduced the net cross sectional area of the structure and introduce localized stress concentration. These stress concentrations can cause ply de-lamination.

The design procedure given in [1, 2] is the different way of approach for calculating the value of stress concentration factor. In this paper the method of reducing the stress concentration was given. He proposed that by increasing the thickness around the hole one can reduce the stress concentration. An observed result of this paper also gives the idea of increasing strength of the joint around the hole.J.M.Whitnfy and R, J.Nuismer [2] two related criteria based on stress distribution were presented for predicting the uniaxial tensile strength of laminated composites containing through the thickness discontinuities of a general shape.Hart-Smith.L.J [1,] considered that net tension failure occurs when the bolt diameter is a large fraction of the strip width. This fraction depends on the type of material and lay-up used. Bearing failure occurs predominantly when the bolt diameter is a small fraction of the plate width. This mode of failure leads to an elongation of the hole. Shear-out failure can be regarded as a special case of bearing failure. Cleavage failures are associated with both an inadequate end distance and too few transverse plies. A simple theory to calculate the elastic stress concentration factors at loaded holes was used. Combining this analysis with tests on composite materials to account for the stress concentration relief that occurs in laminates prior to failure, a considerable generalization to geometries for which data was not available, was achieved here.

1.1Failure Modes in Mechanical Joints

The composite designer must consider four relevant stresses in mechanical joints. The bearing stress,,is the load, P divided by the projected transverse cross sectional area of the hole , The shear-out stress is determined by the longitudinal shear surfaces, and is given as . The net section stress is. , The transverse splitting stress is a localized stress normal to the applied load. When any of these four stresses reach a critical value the joint will fail with a characteristic mode as shown in Fig.2. The gross stress defined as is used to rate the effectiveness of the joint. Joint efficiency is the ratio of the gross section stress at failure to the strength of the laminate in the gross section. Net section failures can be prevented by increasing the ratio of the plate width to hole-diameter, w/d. Generally w/d is sufficient to prevent net section failures. Shear-out failures can be eliminated if the ratio e/d to 3 or greater. Transverse splitting is rare but will occur if there is a high fraction of the fibers in the load direction such as would be the case in a unidirectional composite. Bearing failure is the preferred failure mode since the joined members are not catastrophically separated. Bearing failures are associated with localized hole damage such as local de-lamination and matrix cracking. The Fig.1 shows different bolted joint configurations.

Fig.1 Bolted Joint configurations:

(a)Joint A: (b) Joint B: (c) Joint C

(d) Joint D: (e) Joint E:

Fig.4 Experimental set up for load sharing Analysis.

2 FABRICATIONS AND METHODOLOGY

Composite laminates made of Glass/Epoxy with Hy 971 harder with stacking sequence of 0⁰ / 90⁰ were fabricated by using hand lay-up technique. Symmetrical butt joints with bolts in single row and multiple rows were considered. FEA analysis has been done for different width/diameter ratio (w/d) on failure load was estimated numerically. The main objective of using multi-row joints is to minimize the peak bearing load, avoiding the cut-off due to bearing. To achieve improved strength the joint has to be designed to ensure even load sharing between the fasteners. The experimental set up is shown in Fig.4.

3 LOAD SHARING ANALYSIS

In this present work, the analysis for the determination of the load sharing between individual bolts in a line of bolts in a symmetrical lap joint is outlined.

3.1 Geometry and Model Assumptions

A schematic diagram of the bolted joint is shown in fig 3. The upper part of the diagram shows co-linear bolts in a strap of width w in line with the total applied load P. The lower part of the diagram shows the construction of the joint in which a plate of uniform thickness t_p is situated between two straps of uniform thickness t_s .The assumptions of the model are as follows:

· The ratio of stress to strain is constant.

· The stress is uniformly distributed over the cross sections of main plate and butt straps.

· The effect of friction is negligible

· The bolts fit the holes initially, and the material in the immediate vicinity of the holes is not damaged or Stressed in making the holes or inserting the bolts.

· The relationship between bolt deflection and bolt load is linear in the elastic range.

· Each bolt is assumed to be a short loaded fixed end beam; allowances are made for shearing, bending and bearing effects.

3.2 Model Equations

The relative displacement of plate and straps between the ith and (i+l)th bolts yields for i=l, 2,.... (n-1) is given below:

where R_i is the i^th bolt load and p the external applied load. C_j denotes a constant for the ith bolt dependent upon material properties, plate and strap dimensions and includes effects of bending, shear and bearing. K_p and K_s are given by:

;

where p is the hole spacing or pitch and E_p and E_s denote Young's modulus for the plate and strap respectively. The overall equilibrium condition gives:

;

Each bolt constant C_jhas contributions from the various beam mechanisms in operation as in (2) where

and A_b = л*d²/4, I_b = л*d⁴/64 and d is the bolt diameter. In the above equations, E_bb and E_bbr are Young’s moduli for the bolt under bending and bolt under bearing respectively. E_sbr and E_pbr are the bearing moduli for the strap and plate, assumed equal to the compressive moduli of the respective materials. Assuming that the bolt diameter d is the same for each, then bolts of the same material imply that C_i = C_i+1 = C. It is assumed that: E_pbr = E_p, E_sbr = E_s, E_bbr, E_bb = E_bSolution of these model equations can be got by using any of numerical methods, in this work Gauss Seidel iteration method is used for which the first iteration assumption is Ri=P/n most suitable. The above equation can be re-arranged so that it will be easy to use for GS method in the following way

3.3. Gauss-Siedal Algorithm

Fig. 4 Algorithm for Gauss-Siedal Method

3.4 Result Analysis (Out Put)

1] Enter the values of λ=4.037 and μ= 2.692

Enter the value of load applied in KN: 40

10000.000000 0.000000

23366.376953 4638.946777 920.973938 11079.811523

22302.078125 3689.517822 2747.781494 11266.798828

22113.593750 4203.266113 2525.216553 11164.111328

22215.583984 4077.333984 2518.989258 11194.272461

22190.582031 4096.136230 2523.958008 11195.503906

22194.314453 4094.131104 2522.573730 11195.160156

22193.916016 4094.175537 2522.857422 11195.230469

22193.925781 4094.224121 2522.810547 11195.218750

22193.935547 4094.207031 2522.816406 11195.220703

22193.931641 4094.211182 2522.816406 11195.220703

Fig.5.Relation between Un-evenness factor and bolts at different load

4 OPTIMIZATION OF JOINT EFFICIENCY

For the case of multiple columns of bolts, or a series of loaded holes in an infinitely wide panel, Hart-Smith (1980) suggested that the elastic stress concentration factor K_ic is given by

(1)

Where each bolt is treated individually as a plate of width, w, equal to the pitch, p, Eq. (1) could be modified using the pitch defined in terms of the width, as follows:

where n=number bolts in a row

Therefore (1) can be rewritten as follows:

Where θ is defined as

The Structural efficiency for composite materials of a bolted connection is given by the equation

Efficiency (η) =

where C is correction factor.

Based on the least-squares regression analysis, the correlation coefficients C for the five different configurations with zero degree orientation with respect to the applied load ,as per Hassan 1995 are as bellow

For connection type A, C=0.22

For connection type B, C=0.40

For connection type C, C=0.16

For connection type D, C=0.50

For connection type E, C=0.30

In this analysis C Types of joints were considered

By using above formulae efficiency of the joints was predicted. Variation of the efficiency with respect to the d/w and the ratio of e/d was plotted on the graphs shown in Fig.6 and Fig.7. Table 1 shows various parameters considered for the computation of efficiency. From these graphs we can predict the optimal values of the geometric parameters (like hole dia, width and edge distance) can be determined.

TABLE- 1 Estimation of Efficiency

When e/d=2 and d=6mm

d/w	e/w	θ	w/d	K_ie	K_ic	η
0.1	0.2	-1	10	12.22727	3.47	0.259365
0.15	0.3	-0.166666	6.666666	7.851449	2.507318	0.339007
0.2	0.4	0.25	5	5.75	2.045	0.391198
0.25	0.5	0.5	4	4.55	1.781	0.421111
0.3	0.6	0.666666	3.333333	3.794871	1.614871	0.433470
0.35	0.7	0.785714	2.857142	3.289682	1.503730	0.432258
0.4	0.8	0.875	2.5	2.9375	1.42625	0.420683

When e/d=3 and d=6mm

d/w	e/w	θ	w/d	K_ie	K_ic	η
0.1	0.3	-0.166666	10	11.20454	3.245	0.277349
0.15	0.45	0.388888	6.666666	7.235507	2.371811	0.358375
0.2	0.6	0.666666	5	5.333333	1.953333	0.409556
0.25	0.75	0.833333	4	4.25	1.715	0.437317
0.3	0.9	0.944444	3.333333	3.570512	1.565512	0.447137
0.35	1.05	1	2.857142	3.134920	1.469682	0.442272
0.4	1.2	1	2.5	2.857142	1.408571	0.425963

When e/d=5 and d=6mm

d/w	e/w	θ	w/d	K_ie	K_ic	η
0.1	0.5	0.5	10	10.38636	3.065	0.293637
0.15	0.75	0.833333	6.666666	6.742753	2.263405	0.375540
0.2	1	1	5	5	1.88	0.425531
0.25	1.25	1	4	4.1	1.682	0.445897
0.3	1.5	1	3.333333	3.525641	1.555641	0.449975
0.35	1.75	1	2.857142	3.134920	1.469682	0.442272
0.4	2	1	2.5	2.857142	1.408571	0.425963

Fig.6 Efficiency vs. d/w ratio Fig.7 K_ic vs_.d/w ratio

5 RESULTS AND DISCUSSION

Load shared by each bolt obtained from numerical methods by considering possible failure in the model equations have good agreement with experimentally determined data. Hence the method can be used as a design purposes for a composite laminate that is designed to fail in net tension. The minimum fastener spacing has been validated.
Optimal value of edge to dia (e/d) and width to dia (w/d) ratios were obtained for maximum efficiency.Net tension is prevented at e/d=3 and shear out failure is prevented at w/d=6.
It is observed that un-evenness factor will be same for all applied loads.

6 REFERENCES

1. Hart-Smith J., "Mechanically fastened joints for advanced composites, phenomenological considerations and simple analyses", Fibrous composites in structural design, Plenium Press, (1993).

2. Whitney J. and Nuismer R., "Stress fracture criteria for laminated composites containing stress concentrations", J. Composite Materials, 8,253-265, (1974).

3. Sims G. D., Payned D. R. and Ferriss D. H., "Analysis and experimental validation of structural element test methods", ECCM-7/CTS3, 73-78, Volume 2, (1996).

4. J Niklewicz, D. H. Ferriss, G. J. Nunn and G. D. Sims, “The Use Of Pin Bearing Data For The Preliminary Design Of "Bolted" Joints” CMMT(MN)052,1999.

Technical College - Bourgas,