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A composite double-lap joint with staggered bolts: an experimental and analytical investigation
Composite Structures 54 (2001) 3±15
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A composite double-lap joint with staggered bolts: an experimental and analytical investigation J. Hanauska, V. Kradinov, E. Madenci * Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
Abstract The objective of this study is to investigate the bolt load distribution in double-lap joints with ®ve staggered bolts. The measured strain values near the bolts are used to establish the validity of the analytical predictions based on the concept of virtual work in conjunction with the complex potential theory. The total potential energy also includes the strain energy of the bolts based on a shear deformable beam theory. This method provides the contact stresses around the bolts holes, and thus the loads exerted by each bolt, while accounting for the interaction of bolts. The measured and predicted strain values are in agreement, verifying the accuracy of the predicted bolt loads. Damage and its growth are also examined by applying X-rays to the specimens prior to and after loading. The extent of the damage correlates to higher contact stresses. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Multiple bolts; Composite; Experimental; Analytical
1. Introduction Load transfer among aerospace components made of composite materials is commonly achieved by means of bolted joints. High stress concentrations in the joint area arising from the presence of bolt holes may lead to failure modes of net-section, shear-out and bearing, or a combination. The ability to predict the stress state around bolt holes in lap joints of composite materials is essential to the safe design of structures. The stress distribution around a single pin-loaded hole in a composite material has been studied extensively in the literature. The ®nite element method (FEM) has been utilized to solve both the two- and the threedimensional contact problem between the pin and the hole boundary. Wong and Matthews [1] investigated the eects of hole edge distance on the bearing strength of single- and two-hole bolted joints of glass and carbon ®ber laminates by performing a two-dimensional ®nite element analysis using plane stress elements. Eriksson [2] studied the eects of stacking sequence, friction clearance, bolt stiness and the magnitude of the applied load on a single-hole composite specimen using a twodimensional ®nite element model. Serabian and Oplinger [3] employed FEM to investigate both the linear and *
Corresponding author. Tel.: +1-520-621-6113; fax: +1-520-6218191. E-mail address: [email protected] (E. Madenci).
nonlinear response of 0°=90°S pin-loaded laminates. A three-dimensional FE analysis was performed by Barboni et al. [4] to determine the stress distribution around a pin-loaded hole while accounting for stacking sequence and several geometrical parameters. Ganey [5] modeled both the bolt and lap-joint members using three-dimensional solid brick elements while considering the eects of friction and contact pressure for single-lap joints. Based on an incremental restricted variational principle, Chen et al. [6] developed a ®nite element technique to perform a three-dimensional analysis of a composite lap joint with a single bolt. The boundary element method (BEM) is also applicable for solving two- dimensional contact problems. The formulation of this approach for solving both static and dynamic contact problems was presented by Antes and Panagiotopoulos [7]. Man et al. [8±10] implemented an iterative and fully incremental loading technique into a two-dimensional BEM formulation in order to solve the contact problem and applied it to several cases of pin- loaded lugs. Takahashi [11] solved the problem of a pin-loaded hole with the assumption that both the plate and the pin are elastic with or without the presence of friction. The method of computing the stress ®eld for a lap joint with multiple bolts by FEM usually involves determining the load distribution among the bolts ®rst and then applying the resulting load to a single bolt to determine the contact stresses. However, this approach
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does not account for the possible interaction among the fasteners. As an alternative to the FEM and BEM, the boundary collocation technique can be employed to determine the contact stresses and bolt load distribution in single- and multi-fastener mechanical joints of composite materials. A boundary collocation scheme was employed by Rangavittal et al. [12] to investigate the eects of pin ¯exibility on the bearing stresses around a pin-loaded hole in an in®nite body. Three types of pinhole clearances were evaluated. It was found that pin shape in¯uences the stress distribution around a pinloaded hole. Persson and Madenci [13] employed the boundary collocation technique to investigate the loadcarrying ability of dierent-sized elliptical-shaped pins. In the literature, there are essentially no direct analyses of double-lap bolted joints for solid laminates under general loading conditions and appropriate boundary conditions arising from contact phenomenon. Madenci et al. [14,15] extended the boundary collocation technique for single- and double-lap joints with arbitrarily located multiple bolts. A detailed validation and demonstration of their approach, as well as an extensive review of previous analyses, are reported in detail in [14]. Their method determines the contact stresses and contact region, as well as the bolt load distribution, as part of the solution procedure. However, this method fails to provide converged solutions consistently, depending on the number of bolts and their location in relation to each other or to the free boundaries. This method provided converged results for particular con®gurations, but also suered from consistent convergence arising from the explicit partitioning of the domain associated with the boundary collocation method. In order to circumvent this shortcoming, Kradinov et al. [16] developed a method utilizing a variational formulation in conjunction with the complex potential theory. This method provides the bolt load distribution in single- and double-lap joints while accounting for the contact phenomenon and the interaction among the bolts explicitly under bearing and by-pass loading with or without thermal loading. It is an extension of the analysis introduced by Xiong and Poon [17], and it eliminates the requirement of a two-stage analysis and the associated iterative process. The resulting equations are solved in a coupled manner, leading to the contact stresses, contact region, and bolt load distribution. The number of experimental investigations on the behavior of mechanically fastened composite joints is rather limited. Previous studies investigated only lap joints with a single-fastener or multi-fasteners in tandem or parallel. Hyer and Lui [18] employed a photoelastic method to examine the stress distribution in the vicinity of a pin-loaded hole in a glass/epoxy specimen. Tensile tests of single-hole graphite/epoxy test coupons with
varying ply orientations for both a brittle and a toughened epoxy were performed by Eriksson [19]. Experiments concerning single pin-loaded holes have been performed to validate several failure prediction analyses. Hollmann [20] performed double-lap joint experiments to con®rm predictions of the mode of failure for several laminate con®gurations. Acoustic emission probes were utilized by Persson et al. [21] to determine the onset of delamination near a pin-loaded hole. Recently, Starikov [22] investigated the failure mechanisms of two-, four-, and six-bolt, non-staggered, composite double-lap joints under static and fatigue loading. The present investigation concerns the understanding of bolt load distribution and damage initiation among the staggered bolts of double-lap composite joints. Also, the experimental results are used to verify the accuracy of the analysis to compute bolt load distribution while accounting for the interaction among bolts. In the experimental part of this investigation, quasi-isotropic composite specimens are attached to a steel center member with ®ve staggered bolts forming a double-lap joint. The strain near each hole in the joint is measured, and the distribution of bearing load among the bolts is captured. X-ray radiography was performed to record the damage around the holes in the composite specimens. Around the holes, the radiographs indicate the correlation of bearing damage with forces exerted by each bolt. The two-dimensional analysis of the joint by the concept of virtual work in conjunction with the complex potential theory provides the contact stresses, and thus the bolt load prediction. Section 2 describes the experimental investigation. The stress analysis and bolt load predictions are explained in Section 3. The correlation of experimental measurements with the analytical predictions is discussed in Section 4, followed by the ®nal remarks and conclusions. 2. Experimental investigation The experimental investigation involved the fabrication and preparation of the specimens, testing under uniaxial loading, and their non-destructive evaluation. 2.1. Specimen fabrication and preparation The carbon ®ber/epoxy (CF/E) quasi-isotropic laminates were fabricated with Cytec/Fiberite unidirectional pre-preg tape, Hy-E 5377-2A, consisting of intermediate modulus ®bers and toughened epoxy. The laminates have a stacking sequence of 45= 45=0=902S . The orthotropic lamina properties are speci®ed by elastic moduli EL 161,000 MPa and ET 9000 MPa, shear modulus GLT 6100 MPa, and Poisson's ratio
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Fig. 1. Dimensions of steel and composite plates.
mLT 0:26, where L and T are the longitudinal and transverse directions relative to the ®bers. The ®ber orientation is measured from the loading direction. The fabrication and machining process of the specimens, the measurement and data acquisition system, and the procedure for determining the material properties are explained in [23]. The specimens, having planar dimensions of 50:8 mm 152:4 mm, required the drilling of ®ve, 6.35 mm diameter holes. The thickness of the composite specimens are 2.54 mm. The center steel member has a thickness of 6.35 mm. The specimen geometry and hole positions are described in Fig. 1. The clearance between the bolt and hole is approximately 0.1 mm. With these specimens, double-lap joints were assembled as shown in Fig. 2. The composite specimens were clamped to the steel plate with hardened steel shoulder bolts. The bolts were tightened to a ®nger-tight state. Each specimen was equipped with strain gages, whose locations are prescribed as shown in Fig. 3. The washer size, edge distance, and gage size dictated the strain gage location. The assigned numbers identify the holes, and the strain gages are identi®ed by the letters as shown in this ®gure.
controlled MTS/Sintech RENEW servo-mechanical loadframe with a 20-kip loadcell. The crosshead speed was 0.05 in. per minute. Four quasi-isotropic composite specimens as part of two double-lap joints were tested.
2.2. Testing A picture of the test setup is shown in Fig. 4. The uniaxial tensile tests were performed on a Testworks-
Fig. 2. Schematic of assembled double-lap joint.
Fig. 3. Hole numbering, strain gage placement, and lettering for the front composite plate of the joint.
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Fig. 6. Load history of specimen DLJ-II (* X-ray radiograph).
Fig. 4. Double-lap joint in test ®xture.
These two double-lap joints are labeled as DLJ-I and DLJ-II. The loading histories of double-lap joints DLJ-I and DLJ-II are shown in Figs. 5 and 6. Both joints are subjected to loads in increments of approximately 11,000 N. In the case of DLJ-I, the load on the joint was raised to approximately 100 N, and the strain gage values were set to zero through the use of an oset nulling circuit. DLJ-II had an initial load of 500 N ap-
Fig. 5. Load history of specimen DLJ-I (* X-ray radiograph).
plied before the gages were nulled. A loading increment, as de®ned in this study, consisted of raising the load from the initial load with a zero gage reading to the desired values as shown in Figs. 5 and 6. Upon reaching the desired load level, the joint was unloaded and visually inspected. X-ray radiography was performed on both specimens prior to loading. Specimens of DLJ-I were examined with the X-ray technique after the ®nal loading step of 66,700 N. Specimens of DLJ-II were examined with Xrays between each load increment of 11,000 N and at the ®nal load increment of 44,000 N. The maximum applied stress was 264 and 170 MPa for DLJ-I and DLJ-II, respectively. 2.3. Test results The variations of the strain values measured at six locations on the front specimen of DLJ-I as the applied load was increased are shown in Fig. 7. The strains recorded near each hole are shown in Fig. 8. These measurements indicate the presence of expected symmetry in double-lap joints during testing, thus ensuring the validity of load introduction. As observed in Fig. 8, holes 4 and 5, closer to the loaded edge, carry the majority of the load. The increase of strain with increasing load is evident from the initial state up to nearly 50 MPa of applied stress. Above this load level, however, these two holes no longer exhibit a signi®cant increase in strain. This trend was also observed by Starikov [22] for one of the strain-gaged holes in a non-staggered multi-fastener joint. Also observed in Fig. 8 is that the strains becomes nearly constant near holes 4 and 5 as the strains in the vicinity of holes 1 and 2 begin to increase in a nearly linear manner. As the applied load approaches 150 MPa, strain gage E near hole 5 shows evidence of
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Fig. 7. Applied far-®eld stress versus strain for DLJ-I loaded up to 44 kN.
Fig. 9. Strains near the ®ve holes after loading to 11 kN.
bearing failure. Also at this same load level, it is noted that the strain measured through gage C near hole 3 begins to increase as hole 5 becomes more damaged. The sequence of strain measurements on DLJ-I prior to the bearing failure mentioned above is shown in Figs. 9±11. Examining the measurements over the ®rst loading increment (11 kN) in Fig. 9, it is noted that initially the strain at gage D (hole 4) goes into the tensile region. This is possibly due to a small amount of initial clear-
ance between the bolt and the hole. As the load continues to increase, the curve parallels that of gage E (hole 5). Gages A and B (holes 1 and 2) show no increase in strain through the ®rst loading increment. Gage C (hole 3), however, shows a slight linear increase in strain toward the end of the load step. During the second load increment of 22 kN, it is evident in Fig. 10 that holes 1 and 2 begin to carry a portion of the load. Between approximately 30 and 40
Fig. 8. Strains near the ®ve holes in DLJ-I for a maximum applied load of 44 kN.
Fig. 10. Strains near the ®ve holes after loading to 22 kN.
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Fig. 11. Strains near the ®ve holes after loading to 33 kN.
Fig. 12. First loading to 44 kN after hole 5 becomes damaged.
MPa, the stress±strain curves of holes 4 and 5 start to show less increase in strain with increasing stress. At about 40 MPa, the curve representing hole 3 increases at a new slope, although still in a linear fashion, up to the end of the second load increment. Around 50 MPa the strains ahead of holes 1 and 2 also begin to increase linearly, with approximately the same proportion to each other, until the end of the load increment. By the third load increment of 33 kN, it is apparent from Fig. 11 that the strains at holes 4 and 5 have become nearly constant and holes 1±3 begin to carry more of the load. From about 40 MPa until the highest stress during this load increment, the strains near holes 1 and 2 appear to increase linearly and in a similar relation to each other. The curve depicting hole 3 begins to show similar characteristics to those of holes 4 and 5, i.e., a reduction in strain with increasing stress. By the fourth load increment, up to 44 kN, the trend of load transfer among the bolts is quite evident, as shown originally in Fig. 8. Following the failure of hole 5 and subsequently hole 4 (though not as abrupt a failure), DLJ-I was loaded up to 44 kN and unloaded three separate times. Figs. 12±14 indicate that the load shifted to the three remaining holes as holes 4 and 5 became increasingly damaged. By the last loading to 44.4 kN, the measured strain near hole 3 exhibits the same type of constant strain behavior as that shown by holes 4 and 5 previously. Examining the ®nal loading of DLJ-I to approximately 67 kN, the strains at gages D and E are in the tensile region as shown in Fig. 15, and thus no longer supporting any bolt load. It is also apparent that the center hole transfers the load to the bolts near the free edge (holes 1 and 2) at around 125 MPa.
Due to limitations of the testing equipment, no higher loads were applied. An explanation for the sequence of loading distribution observed in these ®gures may be hole/bolt mismatch, i.e., the clearances between holes 4 and 5 and their bolts was less than that between holes 1±3 and their respective bolts. As the load was increased, the holes with less clearance started carrying the load ®rst and, as these holes became damaged and deformed, the other holes came into contact and also started carrying some of the applied load.
Fig. 13. Second loading to 44 kN after hole 5 becomes damaged.
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Fig. 14. Third loading to 44 kN after hole 5 becomes damaged.
the nature of the scanning process, but the damage was still visible. An X-ray radiograph of the front plate of DLJ-I after having been loaded to approximately 67 kN is shown in Fig. 16. Large delaminations around holes 4 and 5 are visible. Examination of holes 4 and 5 indicates the evidence of cracks, oriented at 45° to the loading direction, emanating from the holes. The damage of specimen of DLJ-I coincides with the data recorded by the strain gages in Fig. 15. Fig. 17 shows an X-ray radiograph of the back of specimen DLJ-II, with close-up views of hole number 5. At the ®rst load step of 11 kN, there is no visible damage. After the load is increased to 22 kN, delaminations begin to grow around the loaded edge of the hole. A small crack is barely visible at approximately 45° from the loading direction. When the load is increased to 33 kN, the delamination and the crack both continue to grow. Up to the ®nal loading of 44 kN, the delamination continued to grow while the crack remained nearly the same length. Also evident in the last X-ray is a small amount of crushing along the loaded edge of the hole. Crushing of the laminate causes material to be pushed up in a direction out of the plane of the laminate, thus creating a thicker area in that region and a lighter area on the X-ray negative. Hole 4 of the same laminate showed similar characteristics but to a lesser degree, which agrees with the strain gage measurements as shown in Fig. 18.
Fig. 15. Application of 67 kN load to DLJ-I after failure of holes 4 and 5.
2.4. Inspection of specimens Application of X-ray radiography to the composite specimen can capture internal specimen damage, such as delamination and crushing. Radio opaque dye was applied to an open edge of the damaged region and the specimen, and X-ray radiography was performed. The X-rays ®lms were then scanned and converted to computer image ®les through the use of a ¯atbed scanner. Some resolution of the images was lost due to
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Fig. 16. Damage to DLJ-I after loading to 66.7 kN.
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Fig. 17. Damage to hole 5 of the back plate of DLJ-II after four load steps.
3. Stress analysis and bolt load prediction An experimental investigation provides only the strain measurements and the sequence of load sharing among the bolts. The load exerted by each bolt, and the stress and strain ®elds in the ®ve-bolt composite doublelap joint are determined by applying the concept of virtual work in conjunction with the complex potential theory. The governing equations are derived by requiring the ®rst variation of the total potential energy to vanish. The in-plane equilibrium equations in each plate are satis®ed exactly by employing complex potential functions in the form suggested by Lekhnitskii [24]. However, each of the bolt equilibrium equations and the boundary conditions are satis®ed by minimizing the total potential energy. Contact between the bolts and the laminates is established by enforcing displacement continuity along the contact region between the bolts and the plates. This is established by incorporating the
work done by the unknown contact forces over the contact displacements into the total potential energy expression. The contact displacements are de®ned by constraint equations that take into account the gap between the bolts and plates. Contact forces are assumed in the form of a trigonometric series that satis®es stressfree conditions at the ends of the contact regions. Since the contact regions are unknown a priori, an iterative scheme is adopted in order to determine the beginning and end angles of the contact regions. Starting with an initial guess, the system matrix is generated to solve for unknown plate and bolt displacements and contact forces simultaneously. The iterative scheme is continued until a con®guration for contact regions is reached where all the contact forces become compressive. The details of this approach are given by Kradinov et al. [16]. As the double-lap joint is subjected to uniform stress, r0 , along the vertical edges, the contact region, whose
J. Hanauska et al. / Composite Structures 54 (2001) 3±15
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For a rigid bolt with radius a, the boundary conditions along the hole boundary can be stated as ur a; h d cos h 1 W2 6 h 6 W1 ; uh a; h d sin h rrr a; h rrh a; h 0;
Fig. 18. Strains near the ®ve holes in DLJ-II for a maximum applied load of 44 kN.
extent is not known, develops between the bolts and the hole boundaries as shown in Fig. 19. The angles W1 and W2 de®ne the extent of the contact region. The bolt exerts loading on the boundary through this contact region. The clearance, k; between the bolt and the hole is assumed to be zero, and the bolt is as considered rigid because bolt ¯exibility is not a signi®cant variable in determining the contact stresses [25]. The coecient of friction between the bolt and the composite is also assumed to be zero.
Fig. 19. Contact region between bolt and hole.
W1 6 h 6
W2 :
2
The components of the displacement ®eld referenced to the polar coordinates r; h are denoted by ur and uh , and those of the stress ®elds by rrr , rhh , and rrh . As shown in Fig. 19, the origins of both of the Cartesian coordinate systems x; y and x0 ; y 0 and the polar coordinate system r; h coincide with the center of the bolt hole. The angles W1 and W2 , as well as the pin displacement d; are determined as part of the solution by imposing the global equilibrium equations and rrr a; h < 0 along the contact region for which rrr a; W1 rrr a; W2 0. The primed coordinate system, x0 ; y 0 , is rotated by an angle of h0 from the x-axis. This angle is de®ned by the pin displacements both in the x- and y-directions, dx and dy , respectively, as dy h0 tan 1 : 3 dx The pin displacement in the x0 direction is given by q 4 d d2x d2y : Along the external boundaries, the composite specimen is subjected to uniform stress along the right vertical edge, and the steel plate is subjected to the displacement constraint along the left vertical edge. The remaining edges are traction free. The dimensions of the composite specimens and the steel plate are shown in Fig. 20. All calculations for the determination of these parameters were performed for an applied stress level of 100 MPa. Because of the nonlinear nature of the load transfer through the unknown contact regions, the analytical predictions were performed at load levels of 11, 22, 33 and 44 kN, corresponding to the applied loads during testing. The strain components were computed at the center locations of the strain gages. The computed and measured values may dier signi®cantly from each other, especially in the presence of sharp strain gradients close to the hole, because the strain is measured as the average strain over a 1 1:5 mm2 area. As illustrated in Fig. 21, the computed strain in the direction of loading exhibits sharp gradients along a line going through the centers of holes 1 and 4. The variations of the radial and circumferential stresses along the hole boundaries are illustrated in Figs. 22 and 23, respectively. These stress components are normalized with respect to the maximum applied stress, r0 170 MPa. As shown in Fig. 22, the
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Fig. 20. Single-lap joint model and schematic discretization.
bearing stresses are lower for holes 1±3 than those close to the loaded edge of the specimen. These results also con®rm the presence of symmetry about the
center hole. Integrating the stress components over each contact region yields the force exerted by each bolt, and thus the prediction of load distribution among the bolts. These predictions are presented in Table 1.
Fig. 21. Variation of the predicted normal strain in the loading direction along the dotted line shown in inset.
Fig. 22. Normalized radial stress distribution along the hole boundaries.
J. Hanauska et al. / Composite Structures 54 (2001) 3±15
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Fig. 23. Normalized tangential stress distribution along the hole boundaries. Table 1 Predicted bolt load distribution Hole/bolt number Percent of total load
1 15.8
2 15.8
3 15.3
4 26.5
5 26.5
Fig. 25. Comparison of measured and predicted strains near holes 1 and 2.
4. Correlation of experimental measurements with analytical predictions In the absence of high strain gradients away from the bolt holes, the computed and measured far-®eld strains, shown in Fig. 24, are in acceptable agreement. A comparison of the measured and predicted strain values near holes 1 and 2 is presented in Fig. 25. As indicated by the measured strain values, holes 1 and 2 show no signs of carrying load, possibly due to bolt-hole clearance, until the uniform applied stress reaches about 40 MPa.
Fig. 26. Comparison of measured and predicted strains near hole 3.
Fig. 24. Comparison of measured and predicted far-®eld strains.
Therefore, the origin of the predicted strain values, corresponding to zero strain, begins at this load level. There exists an acceptable correlation between the measured and calculated strains near holes 1 and 2 after the applied stress reaches approximately 40 MPa. A comparison of the measured and predicted strain values near hole 3 is shown in Fig. 26. From zero applied load to approximately 40 MPa, the measured and predicted strains values appear to be in agreement with each other. As the load is increased above 40 MPa, load transfer to hole 3 due to damage at holes 4 and 5 causes a deviation
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the fact that the strains were measured over an area while the calculations were made at speci®c points. References
Fig. 27. Comparison of measured and predicted strains near holes 4 and 5.
between the measured and predicted strains near hole 3. Because holes 4 and 5 are subjected to very high stresses, thus experiencing damage in the contact region at a very low load level, the correlations of the measured and calculated strain are not acceptable after the initial application of the load, as shown in Fig. 27. Another reason may be due to the highly nonlinear load±strain behavior close to the hole boundary, even at low loads. Insucient bonding between the gages and the laminate may also add to the discrepancy. Gage misalignment and transverse eects were not accounted for in the present analysis. 5. Conclusions The analysis agrees well with the measured data when the measured bearing strain behaves in a linear manner. The large discrepancy between the calculated and the measured strains is largely due to the highly nonlinear nature of the problem. The transfer of loads between the bolts as the holes become damaged is not captured in the analysis. The analysis predicts that the holes close to the loaded edge of the composite plate carry the majority of the load. This prediction is in agreement with both the strain gage readings and the X-ray radiographs. Comparison of the experimental strain measurements with the analytical predictions established the accuracy of the method. The correlation of the global strains showed remarkable agreement. The discrepancy observed between strain measurements and predictions close to the hole boundaries may have been caused by
[1] Wong CMS, Matthews FL. A ®nite element analysis of single and two-hole bolted joints in ®ber reinforced plastics. J Compos Mater 1981;15:481±91. [2] Eriksson IL. Contact stresses in bolted joints of composite laminates. Compos Struct 1986;6:57±75. [3] Serabian SM, Oplinger DW. An experimental and ®nite element investigation into the mechanical response of 0/90 pin-loaded laminates. J Compos Mater 1987;21:631±49. [4] Barboni R, Gaudenzi P, Carlini SA. Three-dimensional analysis of edge eects in composite laminates with circular holes. Compos Struct 1990;15:115±36. [5] Ganey RP. Three-dimensional ®nite element analysis of bolted composite lap joints. Report 93-13, Department of Lightweight Structures, Royal Institute of Technology, Stockholm, Sweden, 1993. [6] Chen W-H, Lee S-S, Yeh J-T. Three-dimensional contact stress analysis of a composite laminate with bolted joint. Compos Struct 1995;30:287±97. [7] Antes H, Panagiotopoulos PD. The boundary integral approach to static and dynamic contact problems. International series of numerical mathematics, ISNM, vol. 10. Boston: Birkhauser; 1992. [8] Man KW, Aliabadi MH, Rooke DP. BEM frictional contact analysis: load incremental technique. Comput Struct 1993;47:893± 905. [9] Man KW, Aliabadi MH, Rooke DP. Stress intensity factors in the presence of contact stresses. Eng Fract Mech 1995;51:591±601. [10] Man KW. Contact mechanics using boundary elements. Topics in engineering, vol. 22. Boston: Computational Mechanics Publishing; 1994. [11] Takahashi S. Elastic contact analysis by boundary elements. Lecture notes in engineering, vol. 67. Berlin: Springer; 1991. [12] Rangavittal HK, Naidu ACB, Dattaguru B, Ramamurthy TS. Analytical solutions for load transfer through smooth elastic pin in an in®nite orthotropic plate. Compos Struct 1995;30: 329±39. [13] Persson E, Madenci E. Composite laminates with elliptical pinloaded holes. Eng Fract Mech 1998;61:279±95. [14] Madenci E, Shkarayev S, Sergeev B, Oplinger DW, Shyprykevich P. Analysis of composite laminates with multiple fasteners. Int J Solids Struct 1998;35:1793±811. [15] Madenci E, Sergeev B, Shipilov Y, Scotti S. Analysis of doublelap composite joints with multiple-row bolts under combined mechanical and thermal loading. In: Proceedings of the International Conference on Joining and Repair of Plastics and Composites. London, England: The Institution of Mechanical Engineers; 1999. p. 55±64. [16] Kradinov V, Barut A, Madenci E, Damodar AP. Analysis of single- and double-lap composite joints with multiple-row bolts under combined mechanical and thermal loading. Int J Solids Struct [in press]. [17] Xiong Y, Poon C. A design model for composite joints with multiple fasteners. Aeronautical note IAR-AN-80, NRC no. 3216, National Research Council, Canada, 1994. [18] Hyer MW, Lui D. Stresses in a quasi-isotropic pin-loaded connector using photoelasticity. Exp Mech 1994;24:48±53. [19] Eriksson I. On the bearing strength of bolted graphite/epoxy laminates. J Compos Mater 1990;24:1246±69. [20] Hollmann K. Failure analysis of bolted composite joints exhibiting in-plane failure modes. J Compos Mater 1996;30:358±83.
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[23] Hanauska J. An experimental and analytical investigation of a staggered fastener composite double-lap joint. MS Thesis, University of Arizona, Tucson, AZ, 1999. [24] Lekhnitskii SG. Anisotropic plates. New York: Gordon and Breach; 1968. [25] Hyer MW, Klang EC. Contact stresses in pin-loaded orthotropic plates. Int J Solids Struct 1985;21:957±75.
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A composite double-lap joint with staggered bolts: an experimental and analytical investigation J. Hanauska, V. Kradinov, E. Madenci * Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
Abstract The objective of this study is to investigate the bolt load distribution in double-lap joints with ®ve staggered bolts. The measured strain values near the bolts are used to establish the validity of the analytical predictions based on the concept of virtual work in conjunction with the complex potential theory. The total potential energy also includes the strain energy of the bolts based on a shear deformable beam theory. This method provides the contact stresses around the bolts holes, and thus the loads exerted by each bolt, while accounting for the interaction of bolts. The measured and predicted strain values are in agreement, verifying the accuracy of the predicted bolt loads. Damage and its growth are also examined by applying X-rays to the specimens prior to and after loading. The extent of the damage correlates to higher contact stresses. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Multiple bolts; Composite; Experimental; Analytical
1. Introduction Load transfer among aerospace components made of composite materials is commonly achieved by means of bolted joints. High stress concentrations in the joint area arising from the presence of bolt holes may lead to failure modes of net-section, shear-out and bearing, or a combination. The ability to predict the stress state around bolt holes in lap joints of composite materials is essential to the safe design of structures. The stress distribution around a single pin-loaded hole in a composite material has been studied extensively in the literature. The ®nite element method (FEM) has been utilized to solve both the two- and the threedimensional contact problem between the pin and the hole boundary. Wong and Matthews [1] investigated the eects of hole edge distance on the bearing strength of single- and two-hole bolted joints of glass and carbon ®ber laminates by performing a two-dimensional ®nite element analysis using plane stress elements. Eriksson [2] studied the eects of stacking sequence, friction clearance, bolt stiness and the magnitude of the applied load on a single-hole composite specimen using a twodimensional ®nite element model. Serabian and Oplinger [3] employed FEM to investigate both the linear and *
Corresponding author. Tel.: +1-520-621-6113; fax: +1-520-6218191. E-mail address: [email protected] (E. Madenci).
nonlinear response of 0°=90°S pin-loaded laminates. A three-dimensional FE analysis was performed by Barboni et al. [4] to determine the stress distribution around a pin-loaded hole while accounting for stacking sequence and several geometrical parameters. Ganey [5] modeled both the bolt and lap-joint members using three-dimensional solid brick elements while considering the eects of friction and contact pressure for single-lap joints. Based on an incremental restricted variational principle, Chen et al. [6] developed a ®nite element technique to perform a three-dimensional analysis of a composite lap joint with a single bolt. The boundary element method (BEM) is also applicable for solving two- dimensional contact problems. The formulation of this approach for solving both static and dynamic contact problems was presented by Antes and Panagiotopoulos [7]. Man et al. [8±10] implemented an iterative and fully incremental loading technique into a two-dimensional BEM formulation in order to solve the contact problem and applied it to several cases of pin- loaded lugs. Takahashi [11] solved the problem of a pin-loaded hole with the assumption that both the plate and the pin are elastic with or without the presence of friction. The method of computing the stress ®eld for a lap joint with multiple bolts by FEM usually involves determining the load distribution among the bolts ®rst and then applying the resulting load to a single bolt to determine the contact stresses. However, this approach
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does not account for the possible interaction among the fasteners. As an alternative to the FEM and BEM, the boundary collocation technique can be employed to determine the contact stresses and bolt load distribution in single- and multi-fastener mechanical joints of composite materials. A boundary collocation scheme was employed by Rangavittal et al. [12] to investigate the eects of pin ¯exibility on the bearing stresses around a pin-loaded hole in an in®nite body. Three types of pinhole clearances were evaluated. It was found that pin shape in¯uences the stress distribution around a pinloaded hole. Persson and Madenci [13] employed the boundary collocation technique to investigate the loadcarrying ability of dierent-sized elliptical-shaped pins. In the literature, there are essentially no direct analyses of double-lap bolted joints for solid laminates under general loading conditions and appropriate boundary conditions arising from contact phenomenon. Madenci et al. [14,15] extended the boundary collocation technique for single- and double-lap joints with arbitrarily located multiple bolts. A detailed validation and demonstration of their approach, as well as an extensive review of previous analyses, are reported in detail in [14]. Their method determines the contact stresses and contact region, as well as the bolt load distribution, as part of the solution procedure. However, this method fails to provide converged solutions consistently, depending on the number of bolts and their location in relation to each other or to the free boundaries. This method provided converged results for particular con®gurations, but also suered from consistent convergence arising from the explicit partitioning of the domain associated with the boundary collocation method. In order to circumvent this shortcoming, Kradinov et al. [16] developed a method utilizing a variational formulation in conjunction with the complex potential theory. This method provides the bolt load distribution in single- and double-lap joints while accounting for the contact phenomenon and the interaction among the bolts explicitly under bearing and by-pass loading with or without thermal loading. It is an extension of the analysis introduced by Xiong and Poon [17], and it eliminates the requirement of a two-stage analysis and the associated iterative process. The resulting equations are solved in a coupled manner, leading to the contact stresses, contact region, and bolt load distribution. The number of experimental investigations on the behavior of mechanically fastened composite joints is rather limited. Previous studies investigated only lap joints with a single-fastener or multi-fasteners in tandem or parallel. Hyer and Lui [18] employed a photoelastic method to examine the stress distribution in the vicinity of a pin-loaded hole in a glass/epoxy specimen. Tensile tests of single-hole graphite/epoxy test coupons with
varying ply orientations for both a brittle and a toughened epoxy were performed by Eriksson [19]. Experiments concerning single pin-loaded holes have been performed to validate several failure prediction analyses. Hollmann [20] performed double-lap joint experiments to con®rm predictions of the mode of failure for several laminate con®gurations. Acoustic emission probes were utilized by Persson et al. [21] to determine the onset of delamination near a pin-loaded hole. Recently, Starikov [22] investigated the failure mechanisms of two-, four-, and six-bolt, non-staggered, composite double-lap joints under static and fatigue loading. The present investigation concerns the understanding of bolt load distribution and damage initiation among the staggered bolts of double-lap composite joints. Also, the experimental results are used to verify the accuracy of the analysis to compute bolt load distribution while accounting for the interaction among bolts. In the experimental part of this investigation, quasi-isotropic composite specimens are attached to a steel center member with ®ve staggered bolts forming a double-lap joint. The strain near each hole in the joint is measured, and the distribution of bearing load among the bolts is captured. X-ray radiography was performed to record the damage around the holes in the composite specimens. Around the holes, the radiographs indicate the correlation of bearing damage with forces exerted by each bolt. The two-dimensional analysis of the joint by the concept of virtual work in conjunction with the complex potential theory provides the contact stresses, and thus the bolt load prediction. Section 2 describes the experimental investigation. The stress analysis and bolt load predictions are explained in Section 3. The correlation of experimental measurements with the analytical predictions is discussed in Section 4, followed by the ®nal remarks and conclusions. 2. Experimental investigation The experimental investigation involved the fabrication and preparation of the specimens, testing under uniaxial loading, and their non-destructive evaluation. 2.1. Specimen fabrication and preparation The carbon ®ber/epoxy (CF/E) quasi-isotropic laminates were fabricated with Cytec/Fiberite unidirectional pre-preg tape, Hy-E 5377-2A, consisting of intermediate modulus ®bers and toughened epoxy. The laminates have a stacking sequence of 45= 45=0=902S . The orthotropic lamina properties are speci®ed by elastic moduli EL 161,000 MPa and ET 9000 MPa, shear modulus GLT 6100 MPa, and Poisson's ratio
J. Hanauska et al. / Composite Structures 54 (2001) 3±15
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Fig. 1. Dimensions of steel and composite plates.
mLT 0:26, where L and T are the longitudinal and transverse directions relative to the ®bers. The ®ber orientation is measured from the loading direction. The fabrication and machining process of the specimens, the measurement and data acquisition system, and the procedure for determining the material properties are explained in [23]. The specimens, having planar dimensions of 50:8 mm 152:4 mm, required the drilling of ®ve, 6.35 mm diameter holes. The thickness of the composite specimens are 2.54 mm. The center steel member has a thickness of 6.35 mm. The specimen geometry and hole positions are described in Fig. 1. The clearance between the bolt and hole is approximately 0.1 mm. With these specimens, double-lap joints were assembled as shown in Fig. 2. The composite specimens were clamped to the steel plate with hardened steel shoulder bolts. The bolts were tightened to a ®nger-tight state. Each specimen was equipped with strain gages, whose locations are prescribed as shown in Fig. 3. The washer size, edge distance, and gage size dictated the strain gage location. The assigned numbers identify the holes, and the strain gages are identi®ed by the letters as shown in this ®gure.
controlled MTS/Sintech RENEW servo-mechanical loadframe with a 20-kip loadcell. The crosshead speed was 0.05 in. per minute. Four quasi-isotropic composite specimens as part of two double-lap joints were tested.
2.2. Testing A picture of the test setup is shown in Fig. 4. The uniaxial tensile tests were performed on a Testworks-
Fig. 2. Schematic of assembled double-lap joint.
Fig. 3. Hole numbering, strain gage placement, and lettering for the front composite plate of the joint.
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Fig. 6. Load history of specimen DLJ-II (* X-ray radiograph).
Fig. 4. Double-lap joint in test ®xture.
These two double-lap joints are labeled as DLJ-I and DLJ-II. The loading histories of double-lap joints DLJ-I and DLJ-II are shown in Figs. 5 and 6. Both joints are subjected to loads in increments of approximately 11,000 N. In the case of DLJ-I, the load on the joint was raised to approximately 100 N, and the strain gage values were set to zero through the use of an oset nulling circuit. DLJ-II had an initial load of 500 N ap-
Fig. 5. Load history of specimen DLJ-I (* X-ray radiograph).
plied before the gages were nulled. A loading increment, as de®ned in this study, consisted of raising the load from the initial load with a zero gage reading to the desired values as shown in Figs. 5 and 6. Upon reaching the desired load level, the joint was unloaded and visually inspected. X-ray radiography was performed on both specimens prior to loading. Specimens of DLJ-I were examined with the X-ray technique after the ®nal loading step of 66,700 N. Specimens of DLJ-II were examined with Xrays between each load increment of 11,000 N and at the ®nal load increment of 44,000 N. The maximum applied stress was 264 and 170 MPa for DLJ-I and DLJ-II, respectively. 2.3. Test results The variations of the strain values measured at six locations on the front specimen of DLJ-I as the applied load was increased are shown in Fig. 7. The strains recorded near each hole are shown in Fig. 8. These measurements indicate the presence of expected symmetry in double-lap joints during testing, thus ensuring the validity of load introduction. As observed in Fig. 8, holes 4 and 5, closer to the loaded edge, carry the majority of the load. The increase of strain with increasing load is evident from the initial state up to nearly 50 MPa of applied stress. Above this load level, however, these two holes no longer exhibit a signi®cant increase in strain. This trend was also observed by Starikov [22] for one of the strain-gaged holes in a non-staggered multi-fastener joint. Also observed in Fig. 8 is that the strains becomes nearly constant near holes 4 and 5 as the strains in the vicinity of holes 1 and 2 begin to increase in a nearly linear manner. As the applied load approaches 150 MPa, strain gage E near hole 5 shows evidence of
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Fig. 7. Applied far-®eld stress versus strain for DLJ-I loaded up to 44 kN.
Fig. 9. Strains near the ®ve holes after loading to 11 kN.
bearing failure. Also at this same load level, it is noted that the strain measured through gage C near hole 3 begins to increase as hole 5 becomes more damaged. The sequence of strain measurements on DLJ-I prior to the bearing failure mentioned above is shown in Figs. 9±11. Examining the measurements over the ®rst loading increment (11 kN) in Fig. 9, it is noted that initially the strain at gage D (hole 4) goes into the tensile region. This is possibly due to a small amount of initial clear-
ance between the bolt and the hole. As the load continues to increase, the curve parallels that of gage E (hole 5). Gages A and B (holes 1 and 2) show no increase in strain through the ®rst loading increment. Gage C (hole 3), however, shows a slight linear increase in strain toward the end of the load step. During the second load increment of 22 kN, it is evident in Fig. 10 that holes 1 and 2 begin to carry a portion of the load. Between approximately 30 and 40
Fig. 8. Strains near the ®ve holes in DLJ-I for a maximum applied load of 44 kN.
Fig. 10. Strains near the ®ve holes after loading to 22 kN.
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Fig. 11. Strains near the ®ve holes after loading to 33 kN.
Fig. 12. First loading to 44 kN after hole 5 becomes damaged.
MPa, the stress±strain curves of holes 4 and 5 start to show less increase in strain with increasing stress. At about 40 MPa, the curve representing hole 3 increases at a new slope, although still in a linear fashion, up to the end of the second load increment. Around 50 MPa the strains ahead of holes 1 and 2 also begin to increase linearly, with approximately the same proportion to each other, until the end of the load increment. By the third load increment of 33 kN, it is apparent from Fig. 11 that the strains at holes 4 and 5 have become nearly constant and holes 1±3 begin to carry more of the load. From about 40 MPa until the highest stress during this load increment, the strains near holes 1 and 2 appear to increase linearly and in a similar relation to each other. The curve depicting hole 3 begins to show similar characteristics to those of holes 4 and 5, i.e., a reduction in strain with increasing stress. By the fourth load increment, up to 44 kN, the trend of load transfer among the bolts is quite evident, as shown originally in Fig. 8. Following the failure of hole 5 and subsequently hole 4 (though not as abrupt a failure), DLJ-I was loaded up to 44 kN and unloaded three separate times. Figs. 12±14 indicate that the load shifted to the three remaining holes as holes 4 and 5 became increasingly damaged. By the last loading to 44.4 kN, the measured strain near hole 3 exhibits the same type of constant strain behavior as that shown by holes 4 and 5 previously. Examining the ®nal loading of DLJ-I to approximately 67 kN, the strains at gages D and E are in the tensile region as shown in Fig. 15, and thus no longer supporting any bolt load. It is also apparent that the center hole transfers the load to the bolts near the free edge (holes 1 and 2) at around 125 MPa.
Due to limitations of the testing equipment, no higher loads were applied. An explanation for the sequence of loading distribution observed in these ®gures may be hole/bolt mismatch, i.e., the clearances between holes 4 and 5 and their bolts was less than that between holes 1±3 and their respective bolts. As the load was increased, the holes with less clearance started carrying the load ®rst and, as these holes became damaged and deformed, the other holes came into contact and also started carrying some of the applied load.
Fig. 13. Second loading to 44 kN after hole 5 becomes damaged.
J. Hanauska et al. / Composite Structures 54 (2001) 3±15
Fig. 14. Third loading to 44 kN after hole 5 becomes damaged.
the nature of the scanning process, but the damage was still visible. An X-ray radiograph of the front plate of DLJ-I after having been loaded to approximately 67 kN is shown in Fig. 16. Large delaminations around holes 4 and 5 are visible. Examination of holes 4 and 5 indicates the evidence of cracks, oriented at 45° to the loading direction, emanating from the holes. The damage of specimen of DLJ-I coincides with the data recorded by the strain gages in Fig. 15. Fig. 17 shows an X-ray radiograph of the back of specimen DLJ-II, with close-up views of hole number 5. At the ®rst load step of 11 kN, there is no visible damage. After the load is increased to 22 kN, delaminations begin to grow around the loaded edge of the hole. A small crack is barely visible at approximately 45° from the loading direction. When the load is increased to 33 kN, the delamination and the crack both continue to grow. Up to the ®nal loading of 44 kN, the delamination continued to grow while the crack remained nearly the same length. Also evident in the last X-ray is a small amount of crushing along the loaded edge of the hole. Crushing of the laminate causes material to be pushed up in a direction out of the plane of the laminate, thus creating a thicker area in that region and a lighter area on the X-ray negative. Hole 4 of the same laminate showed similar characteristics but to a lesser degree, which agrees with the strain gage measurements as shown in Fig. 18.
Fig. 15. Application of 67 kN load to DLJ-I after failure of holes 4 and 5.
2.4. Inspection of specimens Application of X-ray radiography to the composite specimen can capture internal specimen damage, such as delamination and crushing. Radio opaque dye was applied to an open edge of the damaged region and the specimen, and X-ray radiography was performed. The X-rays ®lms were then scanned and converted to computer image ®les through the use of a ¯atbed scanner. Some resolution of the images was lost due to
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Fig. 16. Damage to DLJ-I after loading to 66.7 kN.
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Fig. 17. Damage to hole 5 of the back plate of DLJ-II after four load steps.
3. Stress analysis and bolt load prediction An experimental investigation provides only the strain measurements and the sequence of load sharing among the bolts. The load exerted by each bolt, and the stress and strain ®elds in the ®ve-bolt composite doublelap joint are determined by applying the concept of virtual work in conjunction with the complex potential theory. The governing equations are derived by requiring the ®rst variation of the total potential energy to vanish. The in-plane equilibrium equations in each plate are satis®ed exactly by employing complex potential functions in the form suggested by Lekhnitskii [24]. However, each of the bolt equilibrium equations and the boundary conditions are satis®ed by minimizing the total potential energy. Contact between the bolts and the laminates is established by enforcing displacement continuity along the contact region between the bolts and the plates. This is established by incorporating the
work done by the unknown contact forces over the contact displacements into the total potential energy expression. The contact displacements are de®ned by constraint equations that take into account the gap between the bolts and plates. Contact forces are assumed in the form of a trigonometric series that satis®es stressfree conditions at the ends of the contact regions. Since the contact regions are unknown a priori, an iterative scheme is adopted in order to determine the beginning and end angles of the contact regions. Starting with an initial guess, the system matrix is generated to solve for unknown plate and bolt displacements and contact forces simultaneously. The iterative scheme is continued until a con®guration for contact regions is reached where all the contact forces become compressive. The details of this approach are given by Kradinov et al. [16]. As the double-lap joint is subjected to uniform stress, r0 , along the vertical edges, the contact region, whose
J. Hanauska et al. / Composite Structures 54 (2001) 3±15
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For a rigid bolt with radius a, the boundary conditions along the hole boundary can be stated as ur a; h d cos h 1 W2 6 h 6 W1 ; uh a; h d sin h rrr a; h rrh a; h 0;
Fig. 18. Strains near the ®ve holes in DLJ-II for a maximum applied load of 44 kN.
extent is not known, develops between the bolts and the hole boundaries as shown in Fig. 19. The angles W1 and W2 de®ne the extent of the contact region. The bolt exerts loading on the boundary through this contact region. The clearance, k; between the bolt and the hole is assumed to be zero, and the bolt is as considered rigid because bolt ¯exibility is not a signi®cant variable in determining the contact stresses [25]. The coecient of friction between the bolt and the composite is also assumed to be zero.
Fig. 19. Contact region between bolt and hole.
W1 6 h 6
W2 :
2
The components of the displacement ®eld referenced to the polar coordinates r; h are denoted by ur and uh , and those of the stress ®elds by rrr , rhh , and rrh . As shown in Fig. 19, the origins of both of the Cartesian coordinate systems x; y and x0 ; y 0 and the polar coordinate system r; h coincide with the center of the bolt hole. The angles W1 and W2 , as well as the pin displacement d; are determined as part of the solution by imposing the global equilibrium equations and rrr a; h < 0 along the contact region for which rrr a; W1 rrr a; W2 0. The primed coordinate system, x0 ; y 0 , is rotated by an angle of h0 from the x-axis. This angle is de®ned by the pin displacements both in the x- and y-directions, dx and dy , respectively, as dy h0 tan 1 : 3 dx The pin displacement in the x0 direction is given by q 4 d d2x d2y : Along the external boundaries, the composite specimen is subjected to uniform stress along the right vertical edge, and the steel plate is subjected to the displacement constraint along the left vertical edge. The remaining edges are traction free. The dimensions of the composite specimens and the steel plate are shown in Fig. 20. All calculations for the determination of these parameters were performed for an applied stress level of 100 MPa. Because of the nonlinear nature of the load transfer through the unknown contact regions, the analytical predictions were performed at load levels of 11, 22, 33 and 44 kN, corresponding to the applied loads during testing. The strain components were computed at the center locations of the strain gages. The computed and measured values may dier signi®cantly from each other, especially in the presence of sharp strain gradients close to the hole, because the strain is measured as the average strain over a 1 1:5 mm2 area. As illustrated in Fig. 21, the computed strain in the direction of loading exhibits sharp gradients along a line going through the centers of holes 1 and 4. The variations of the radial and circumferential stresses along the hole boundaries are illustrated in Figs. 22 and 23, respectively. These stress components are normalized with respect to the maximum applied stress, r0 170 MPa. As shown in Fig. 22, the
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Fig. 20. Single-lap joint model and schematic discretization.
bearing stresses are lower for holes 1±3 than those close to the loaded edge of the specimen. These results also con®rm the presence of symmetry about the
center hole. Integrating the stress components over each contact region yields the force exerted by each bolt, and thus the prediction of load distribution among the bolts. These predictions are presented in Table 1.
Fig. 21. Variation of the predicted normal strain in the loading direction along the dotted line shown in inset.
Fig. 22. Normalized radial stress distribution along the hole boundaries.
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Fig. 23. Normalized tangential stress distribution along the hole boundaries. Table 1 Predicted bolt load distribution Hole/bolt number Percent of total load
1 15.8
2 15.8
3 15.3
4 26.5
5 26.5
Fig. 25. Comparison of measured and predicted strains near holes 1 and 2.
4. Correlation of experimental measurements with analytical predictions In the absence of high strain gradients away from the bolt holes, the computed and measured far-®eld strains, shown in Fig. 24, are in acceptable agreement. A comparison of the measured and predicted strain values near holes 1 and 2 is presented in Fig. 25. As indicated by the measured strain values, holes 1 and 2 show no signs of carrying load, possibly due to bolt-hole clearance, until the uniform applied stress reaches about 40 MPa.
Fig. 26. Comparison of measured and predicted strains near hole 3.
Fig. 24. Comparison of measured and predicted far-®eld strains.
Therefore, the origin of the predicted strain values, corresponding to zero strain, begins at this load level. There exists an acceptable correlation between the measured and calculated strains near holes 1 and 2 after the applied stress reaches approximately 40 MPa. A comparison of the measured and predicted strain values near hole 3 is shown in Fig. 26. From zero applied load to approximately 40 MPa, the measured and predicted strains values appear to be in agreement with each other. As the load is increased above 40 MPa, load transfer to hole 3 due to damage at holes 4 and 5 causes a deviation
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the fact that the strains were measured over an area while the calculations were made at speci®c points. References
Fig. 27. Comparison of measured and predicted strains near holes 4 and 5.
between the measured and predicted strains near hole 3. Because holes 4 and 5 are subjected to very high stresses, thus experiencing damage in the contact region at a very low load level, the correlations of the measured and calculated strain are not acceptable after the initial application of the load, as shown in Fig. 27. Another reason may be due to the highly nonlinear load±strain behavior close to the hole boundary, even at low loads. Insucient bonding between the gages and the laminate may also add to the discrepancy. Gage misalignment and transverse eects were not accounted for in the present analysis. 5. Conclusions The analysis agrees well with the measured data when the measured bearing strain behaves in a linear manner. The large discrepancy between the calculated and the measured strains is largely due to the highly nonlinear nature of the problem. The transfer of loads between the bolts as the holes become damaged is not captured in the analysis. The analysis predicts that the holes close to the loaded edge of the composite plate carry the majority of the load. This prediction is in agreement with both the strain gage readings and the X-ray radiographs. Comparison of the experimental strain measurements with the analytical predictions established the accuracy of the method. The correlation of the global strains showed remarkable agreement. The discrepancy observed between strain measurements and predictions close to the hole boundaries may have been caused by
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[23] Hanauska J. An experimental and analytical investigation of a staggered fastener composite double-lap joint. MS Thesis, University of Arizona, Tucson, AZ, 1999. [24] Lekhnitskii SG. Anisotropic plates. New York: Gordon and Breach; 1968. [25] Hyer MW, Klang EC. Contact stresses in pin-loaded orthotropic plates. Int J Solids Struct 1985;21:957±75.