Interlaminar Stress Behaviour of the Flywheel Rotor

Interlaminar Stress Behaviour of the Flywheel Rotor 1.1 History of the Flywheel For over thousands of years, flywheels have been used in potter wheels and spindle wheel whorls Gowayed et al., 2002 The concept of storing energy in a rotating disk dates as far back as 2400 BC when rotating wheels were used by Egyptians to handcraft pottery. In fact, flywheel systems were widely used in everyday life (Genta, 1985): in warring chariots, water pumps and even power generations. However, these conventional flywheels are not as efficient as energy storage devices due to the large amount of mass required for the relatively meagre amount of energy stored not to mention the capability of delivering power for only a relatively short period. The advent of the industrial revolution brought about the significant advances of the flywheels. In the 18th century, Man witnessed the widespread use of metal in the construction of machines and soon, flywheel had found its way into steam engines. This development of the flywheel had been attributed to the works of James Watt. With flywheels made of cast iron, a higher mass moment of inertia could be achieve and thus a significant weight saving as well. During the industrial revolution, James Pickard developed a solution for transforming reciprocating to rotary motion with the combination of a crank and flywheel. And it was not till the last thirty years that we witness high performance flywheels being significantly developed with marked improvement and demonstrate the potential as energy storage systems in a wide range of applications. The energy crisis of then 1970s marked the beginning of another significant era for the development of flywheels as the need to search for an alternative energy storage implement. Large amount of money were invested by the governments of many nations into the development of flywheel energy storage technology with subsequent establishment of research programs in the development of flywheel devices as alternative energy storage systems (Genta, 1985). However, development pace since to slow down as fuel prices begin to stabilized in the early 1980s. That particular time period was nevertheless a crucial development era for the flywheel; during which the use of flywheels are explored and developed for electrical vehicles. In addition it was also explored as a device to help utilities manage peak power demand. With the incorporation of high specific strength advanced composite materials into flywheel designs, weight reductions and strength increment can be achieved; and this cannot be achieved with the use of metallic alloy. However, the use of flywheels can yet be commercialized even with the significant improvements in the design concept of flywheel as it remains a challenge still to design flywheel systems that are cost competitive to other energy storage devices. In addition, composites, though stronger than metal, would require the use of advanced bearings due to the inability to withstand certain forces exposed in high performance application (Kim, T.H. 2003). The 1990s witnessed developments in stronger, lightweight composite materials, magnetic bearings and other electronic devices, and all of which contribute to the exciting development of the flywheel. Excessively high rotational speed could now be reached, with a subsequent increment in energy stored, making them a possible candidate again for energy storage system of superior performance. As a summary, the improvement in the flywheel quick energy recovery, high efficiency, low maintenance and long service life, high amount of stored energy per unit volume and mass, high output power levels, as well as lower product and operational cost (Horner, 1996) as well as environmental friendly components are all that have made the flywheel energy storage system a feasible option. 1.1.2 Flywheel as an Energy Storage Device Flywheel energy storage (FES) has, in various past researches, proven its superiority over conventional battery technology based energy storage system in terms of its higher energy density, durability, rapid charge and discharge capability, as well as its tolerance over a wide range of temperature with very minute environmental concerns; and with the progress in power electronics, loss reductions techniques and advanced materials, the then seemingly inconceivable idea of economical flywheel energy storage (FES) devices are no longer chimerical (Hebner et al, 2002). In fact, all of the prepossessing characteristics mentioned briefly above (in comparison with those that of conventional battery system) are what that result in the advanced flywheel systemsà ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒ ¢Ã¢â‚¬Å¾Ã‚ ¢ appeal as one of the fast- gaining attention alternative for energy storage devices. (Arvin Bakis, 2006) A flywheel is an inertial energy storage system where the rotating mass functions as the energy storage retainer. While connected to the motor (possibly electrical motor), a flywheel can be accelerated to a specific angular velocity. In this process, electrical energy was converted into kinetic energy and the rotating inertia of the composite rotor acts to store this form of mechanical energy. When the need arises, the mechanical energy can be transformed back to electrical power by the motor. Thus, the motor in the flywheel system acts not merely as a motor to accelerate the energy storing process but also as a generator in recovering the transformed electrical energy. A typical flywheel energy storage system consists of five primary components (Lazarewicz et al., 2006), namely the rotor, the bearings, the motor/generator unit, the vacuum enclosure and the power electronics. A rotor consists of a hub and rim. As the rim is the main rotating mass of the rotor, it is quite easy to comprehend the fact that the rim acts to store most of the energy whereas the hub functions to connect to the rim to a shaft. The rotating flywheel shaft on the other hand is supported by bearings which could either be of the mechanical or magnetic variety. These bearings allow for low resistance to rotor rotation. However, magnetic bearings are preferred over mechanical ones due to the energy loss associated with energy loss. The motor, as the third major component introduced, acts to accelerate the rotor when electrical energy us supplied to it; whereas the generator acts to extract electrical energy from the rotating rotor by decelerating the rotor. This is in accordance with the principle of conservation of energy. As a result of the rotor deceleration, torque is inevitably produced and is typically transferred between the rotor rim and the motor unit via the hub and the shaft. In addition, as the occurrence of input and output events are not concurrent, the combination of the motor and generator into a single functional unit is typically done to the advantages of weight and cost reduction (Hebner et al. 2002). In addition, a low pressure, vacuum environment is maintained via the use of a pressure vessel enclosure, which serves also to support the structural assemblage of the flywheel and bearing system. This vacuum compartment also serves to house all of the rotating components of the flywheel to reduce aerodynamic drag. Other than that, such a compartment is also crucial in protecting the system from catastrophic failure as a consequence of high energy debris. The power electronics on the other hand act as the interface between the motor/ generator unit and that of the electrical power system by converting the input power into a suitable electrical signal for the operations of the motor/ generator unit. Flywheel energy storage devices have the potential to store a higher amount of energy per unit mass than typical chemical batteries. Where design weight is of major concern, and where maximum energy storage is a fundamental necessity, flywheel energy storage systems seem to offer the most appealing capabilities. This is particularly so in space applications where the crucial design weight necessitates the need for a high energy storage capacity in the smallest available size and mass (and thus weight). Other than its high specific energy density, flywheel also possesses superior specific power and when used with magnetic bearings and advanced motor/ generator system, more than 90% of the storage energy can be retrieved, an efficiency far more superior than that if conventional chemical batteries are used where the reclaimed energy constitutes less than 80% of the energy input. In addition, the increment of the amount of energy stored in flywheels can be achieved via the increase of s peed of the rotating rotor while chemical batteries would possibly need some reassembling of the connections from parallel to serial. Flywheels are very effective devices in avoiding unbalanced or oversized design of power systems due to the way they store energy and this is especially crucial in the deliverance of peak power on demand. For instant access to the desired efficiency of energy storage as well as energy required, flywheels repeat the charging and discharging cycle. This process of charging and discharging occurs at a rapid speed as both processes occur in the very same motor/generator. In addition, the flywheel life will not be affected with the large amount of charging and recharging cycles whereas chemical batteries undergoing a similar process will need a replacement after every few years. In addition, flywheel, as a mechanical type of battery, is also tolerant of the extremity of temperatures and as flywheels do not contain acidic and other hazardous material, flywheels are easily handled during manufacturing process, and disposed of at the end of the flywheelsà ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒ ¢Ã¢â‚¬Å¾Ã‚ ¢ lif e cycles. Flywheel energy storage systems had since found its way into various applications such as transportation and space satellites, to name a few. In transportation, flywheel systemsà ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒ ¢Ã¢â‚¬Å¾Ã‚ ¢ deep recharge and rapid charging capability, the ability to provide high pulses of power as well as the tolerance to a wide operating temperature range as well as the longer operating life on top of weight reduction make flywheel systems an obvious choice in replacing chemical batteries in mobile applications such as electric vehicles. (Hebner et al 2002). A reliable, steady state power quality is of vital importance for critical manufacturing, hospitals, and internet servers. In this context, the flywheel energy storage systems have also found its way into electrical load levelling application such as in ensuring an uninterruptible power supply by providing a smooth and effective transition between a main power source when necessary (Hebner et al 2002) This is one of the current capability of the flywheel and it seems promising that as the technology improves in the near future, flywheel could possibly be applied to peak power managing, where excess energy produced us stored and later released at the peak time in energy consumption. The concept of flywheel is also not novel in the hybrid electric vehicles (HEVs) industry, where small combustion engine is operated while the vehicle is moving at a constant speed. The acceleration process is executed with the extra power provided by the additional battery power supplier. This additional power, on the other hand is generated and stored in the battery when the vehicle brakes such that no additional power is wasted in the form of heat dissipation generated by the friction during brakes. At the present moment, flywheel seems promising in being use for hybrid buses as chemical battery is expensive. In space applications, light weight, compact with high energy density storage capability devices are highly sought after. With increase performance demands on space systems, space programs had had to make consistently huge effort in reducing rim mass to increase payload capacity as well as reduce launch/ fabrication costs. Although chemical batteries had long been a trusted source of energy but flywheel offers much better weight and life benefits as well as the potential to store a larger amount of energy at a lower weight, not to mention the capability to be used as attitude control actuators in replacing reaction flywheel assemblies and control moment gyros. In 2000, Truong et al introduced the Flywheel Energy Storage Demonstration Project, initiated at the NASA Glenn Reasearch Center as a possible replacement for the Battery Energy Storage System on the International Space Station; whereas Fausz et al. had, in the very same year reported that the Flywheel Attitude Control, Energy Transmission and Storage (FACETS) system could combine all or parts of the energy storage, attitude control, and power management and distribution (PMAD) subsystems into a single system, this significantly decreasing flywheel mass (and volume). Thus, in space applications, crucial weight reductions for satellites could, and have hitherto, been achieved with the use of the multi-function high speed flywheel system which not only functions as energy storage but also in providing a gyroscopic effect for attitude control. (Bitterly, 1998, Hebner et al., 2002) But even until recently, the historical development of flywheels and their uses has largely been dependent on advances in both materials and machine technology, coupled with opportunity and necessity (Horner, et al,.1996) However with technological advancements in such a rapid pace, it is not hard to envisage the status of flywheels in the near and distant future. 1.1.3 The Use of Composite and Fiber- Reinforced Materials in Flywheel Design The kinetic energy stored in a flywheel rotor increases linearly with mass but quadratically with rotational speed. . With the increasing demand for high energy storage, flywheels in present applications are often designed for high angular velocities; and these correspond to large centrifugal loads and consequently a higher circumferential and radial stresses, i.e. the dominant stress distribution are hoop stresses (concentric). In this context, the use composite materials with fibers of high unidirectional strength would be desirable. (Shah, 2008) For a fixed axis rotation, the energy stored in a thin rotating ring rotor is [1.3.1] Where I = the rotor moment of inertia ÃÆ' °Ãƒâ€šÃ‚ Ãƒ ¢Ã¢â€š ¬Ã‚ ºÃƒâ€¦Ã‚ ¡ = is the rotor angular velocity. It seems feasible that to increase the stored energy, the mass of the flywheel must be increased and thus its moment of inertia. However, it must be noted that the energy is only linearly proportional to the mass of the flywheel whereas the energy is proportional to the square of the rotational speed. These relations indicate that the rotational speed for a given radius will have a higher influence to the energy density than that of the mass of the flywheel; and to achieve a high rotation speed, a high strength per weight material must be used. Further derivations of the equations below will explain this condition. Resultant force along the hoop and circumferential directions [1.3.2] [1.3.3] Where = force summations in the radial direction = force summations in the circumferential direction dm = mass of the mass element located at radius r rdÃÆ' °Ãƒâ€šÃ‚ Ãƒâ€¦Ã¢â‚¬Å"Æ’ = arc length of the mass element located at radius r ar = radial acceleration of the mass element located at radius r From equation [1.3.3], [1.3.4] With [1.3.5] Where ÃÆ' Ãƒâ€šÃ‚ = mass element density V=mass element velocity B=mass element width Substitute equation [1.3.5] into [1.3.4] and knowing that V=rÃÆ' °Ãƒâ€šÃ‚ Ãƒ ¢Ã¢â€š ¬Ã‚ ºÃƒâ€¦Ã‚ ¡ and for ÃÆ' °Ãƒâ€šÃ‚ Ãƒâ€¦Ã¢â‚¬Å"Æ’ [1.3.6] The tensile stress in the circumferential direction [1.3.7] The stress in a thin- ring rotor is: [1.3.8] It is observed that the maximum speed achievable by a flywheel rotor is limited by the strength of the material from which it is made. The critical speed of the thin ring rotor can be approximated as [1.3.9] Where is the material ultimate strength. From the substitution of equation [1.3.9] into [1.3.1], the specific energy stored in the rim is obtained [1.3.10] K= Flywheel shape factor (Typically 1 for uniform stress disc and 0.5 for thin ring) The dependency of the maximum specific energy stored in the flywheel on the specific strength of the material is thus observed. With the demand for high specific energy in flywheel rotor design, the use of suitable material is thus of paramount importance. Due to their high stiffness to strength, composite materials have successfully been established in flywheel rotor design. Fiber reinforced composites are particularly attractive for use as flywheel materials due to their high strength and low density (Takahashi et al., 2002) The use of composite materials in flywheel designs offer numerous advantages over metallic alloys, including weight and increased strength. This is due to the high tensile strength of the fiber reinforcement phase. In 1986, a composite flywheel rotor was developed by Potter and Medicott for used in vehicle applications. In 1995, the study by Curtiss, et al. shown that the composite Carbon fiber epoxy disc rotor is capable of a 38% higher rim speed or 91% greater rotor energy density than a rotor built of an isotropic high strength to weight ratio Titanium or steel alloys. The carbon fiber reinforced plastic (CFRP) flywheel proposed by Kojima, et al. (1997) shown that high-modulus graphite/epoxy filament wound composite flywheel is able to rotate at a higher speed. The polar woven flywheel by Huang (1999) was shown to possess weight savings features as well as the improvement in life and reliability of the total spacecraft system, and in 2002, the Multi-Direction Composite (MDC) flywheel systems was reported by Gowayed and Flowers. The MDC flywheel system studied employed a new approach to strengthen flywheels with additional reinforcement in the radial direction along with the typical hoop direction reinforcement. In fact, analytical and numerical approaches had over the years been presented to determine the stress, and displacement distribution of the rotor. With the increasing demand for high energy storage, flywheels in present applications are often designed for high angular velocities; and these correspond to large centrifugal loads and consequently a higher circumferential and radial stresses. And the determination of these stresses as well as the ply orientation became especially crucial. As early as 1977, Danfelt et al. published an analytical method for a hybrid multi-rim flywheel with ply-by-ply variation of material properties and based on the assumption of axisymmetry. The method by Danfelt was later extended by Tzeng (1997,2003) which accounts for viscoelasticity effects. In addition, the original method by Danfelt had also been supplemented by a series of researches by Ha with additional consideration of the interference between adjacent rims and varying fiber angles (Ha et al., 1998), the rim radii of numerous material lay- ups for a constant angular velocity (Ha et al. 1999b), residual stresses due to the curing process (Ha et al., 2001) and the subsequent research on a split- type hub (Ha et al., 2006). The effect of rim thicknesses and angular velocity was studied by Arvin and Bakis (2006) while Fabien (2007) studied the optimal continuous variation of fiber angle in a single-material rotor. Other than that, finite element approaches have also been used for stresses and displacement computations which, though computationally more demanding, have gained importance for the analysis and design optimization of flywheel rotors because of the greater modeling depth offer by such methods. It is also possible to assemble the flywheels as a hybrid with rims of different materials in a sequence of increasing ratio of stiffness per density value E/ÃÆ' Ãƒâ€šÃ‚  for increasing radius, r (Arvin Bakis, 2006) using a method called ballasting. From their studies, with circumferential fiber reinforcement, the radial stress distribution is purely tensile with a maximum located approximately in the midplane between the inner and outer radii. But with two-material rotor, the radial stresses turn compressive in the region near the material interface due to the lower stiffness of the inner material which would result in greater expansion. A compressive stress minimum thus exists at the material interface, with two tensile stress maxima found close to the innermost and outermost radius. Despite the increase in circumferential stress level for the outer composite carbon/epoxy rim, such a condition still arises due to the lower radial stresses as a consequent of rotor strength inc rement. After all, composite materials are generally weaker in the transverse direction than in the longitudinal direction. As fiber reinforcement is typically aligned in the circumferential direction, radial tensile stress is often more crucial in comparison with the other mode of stresses due to the weaker strength in this direction. Thus, the dominating stresses are typically those of the circumferential and radial stresses. In this context, much effort had been invested to enhance the efficiency of the composite flywheel rotors by applying stress reduction methods. In view of this method, Danfelt et al. (1997) suggested a sandwich-like rim lay-up with a compliant material between the composite rims of one material to decrease interlaminar stress transmission. 1.2 Literature Review 1.2.1 Interlaminar stresses of Composite Laminates Interlaminar stresses arise when there are discontinuities in the load path, such as free edges and notches. (Wilkins, 1983). In particular, models with a significant amount of curvature. This is because the presence of high interlaminar stresses due to the effect of shell curvature could result in delamination and possibly failure of the laminate at a lower load than that predicted by in-plane failure criteria had they not been properly accounted for. (Edward, K.T., Wilson, R.S. and McLean, S.K. ,1989; Lagace, P.A., 1983) The accurate determination of interlaminar stresses are thus crucial in the design of laminated composite models as the interfacial surfaces of a laminate represent planes of minimum strength (Pagano, N.J. Pipes, R. B., 1973). Classical laminated plate theory (CLPT) was formed in conjunction with the kinetical assumptions of Kirchhoff classical plate theory by assuming a layerwise plane state of stress. However, 2-D CLPT theory alone is not sufficient to explain stress concentration phenomena in various lightweight constructions in aviation vehicle, such as the free-edge effect where full-scale 3-D and singular stress fields occur in the interfaces between two dissimilar layers along the free edges of thermally and/ or mechanically loaded laminates (Mittelstedt Becker, 2003) which decay rapidly with increasing distance from the laminate edge. Such stress localization problems is caused by the discontinuous change of the elastic material properties of the laminate plies at the interfaces and might result in premature failure of the laminate. This is thus an area of concern by designers and much researched has been done since the early 1970s, with the studies initiated by of Pipes and Pagano on the free edg e effects in laminated structures. Early analytic studies were conducted by Hayashi (1967) on edge stress effects consisting of anisotropic plies and adhesive layers transferring interlaminar shear stresses. In early 1970s, Pagano and Pipes also introduced approximation equations for interlaminar normal stresses in the interfaces and was expanded by Conti/ De Paulis in 1985 for the stress- approximation in angle-ply laminates and the calculation of interlaminar stress distribution through the laminate thickness. Whitney simple stress approximations in 1973 did not fulfill the continuity conditions in the interfaces, although Whitney assumption of products of exponential and trigonometric functions did fulfilled the equilibrium conditions and the given traction-free boundary conditions. Researches in the area of free edge effects were also done using various approaches by Tang and Levy (1975) with layerwise series expansion, Hsu and Herakovich (1977) with edge displacement fields in the form of trigonometric and exponential terms, Wang Dickson (1978) with the expansion of the displacement fields into series of Legendre polynomials. However, much discrepancy has been reported. In 1981, series expansions for the stresses in the inner laminate regions and in the vicinity of the free laminate edges by Bar-Yoseph/Pian.CLPT was recovered in the inner laminates with this zero-order approach and unknown parameters obtained by minimizing the laminate complementary potential. The subsequent work by Bar-Joseph used the principle of minimum complementary potential, leading to an eigenvalue problem. The approach used by Bar-Yoseph allowed the continuity of interlaminar stresses in the interfaces as well as the fulfilment of the conditions of traction free surfaces of the laminate. The force balance method by Kassapoglou/Lagace in 1986 and 1987 was developed. Stresses were assumed to consist of layerwise products of in-plane exponential terms and polynomials through the thickness with adjustments done on the thickness terms to satisfy the continuity of all interlaminar stresses in the laminate interfaces and such that they blend into CLPT in the inner laminate regions. Despite its simplicity, the force balance method exhibited good performance even for thick laminates and was thus further explored and refined by other authors. The effects of transverse shear and continuity requirements for both displacements and interlaminar stresses on the composite interface was accounted for by Lu and Liu in developing an Interlaminar Shear Stress Continuity Theory (ISSCT) capable of being used for finite element formulation (Lu, X., Liu, D., 1990). Through that particular theory, interlaminar shear stress could be obtained directly from the constitutive equations. But as the deformation in the thickness direction was neglected during the formulation of the theory, the interlaminar normal stress could not be calculated directly from the constitutive equations. Other than that, a small discrepancy between the results of theirs and that of Pagano elasticity solution in the interlaminar shear stresses small aspect ratios composite laminates was observed (Pagano, N.J. 1969). Although, rigidly bonded laminated composite materials models are always assumed in conventional analysis; but it must be noted that poor bonding and low shear modulus could result in a non-rigid composite interface. As a continuation of the ISSCT, Lu and Liu (1992) later developed the Interlayer Shear Slip Theory (ISST) based on a multilayer approach in investigating the effect of interfacial bonding on the behavior of composite laminates. The Hermite cubic shape functions was used as the interpolation function for composite layer assembly in the thickness direction, and the closed-form solution is obtained for the cases of cylindrical bending of cross-ply laminates with non-rigid interfaces. However, results shown that at some special locations, namely singular points, the transverse shear stress or in-plane normal stress remains insensitive to the condition of interfacial bonding. A closed- form solution was later derived by Lee and Liu (1992) for the complete analysis of interlaminar stresses for both thin and thick composite laminates subjected to sinusoidal distributed loading. The theory was proven to satisfy the continuity of both interlaminar shear stress and interlaminar normal stress at the composite interface and also the interlaminar stresses could be determined directly from the constitutive equations An accurate theory for interlaminar stress analysis should consider the transverse shear effect and continuity requirements for both displacements and interlaminar stresses on the composite interface. It is also advantageous if the formulation is variationally consistent so that it can also be used for finite element formulation. (Kant, T., Swaminathan, 2000) Using the first order shear deformation theory, the interlaminar stresses in laminated composite cylindrical stripes under dynamic loading are studied. Dynamic equations of equilibrium are solved by a combination of Navier approach and a Laplace transform technique. Dynamic magnification factor is calculated for the stresses and deflections for various types of loading and for different values of the geometric parameters. (Bhaskar, K. Varadan, T.K., 1993). Higher order layerwise theorectical framework has been used by Plagianakos and Saravanos (2008) in predicting the static response of thick composite and sandwich composite plates. The displacement field in each discrete layer through the thickness of the laminate include quadratic and cubic polynomial distributions of the in- plane displacements, as well as the linear approximations assumed by linear layerwise theories in addition to the Ritz- type exact solution used to yield the structural response of the thick structure. The formulation has been found to be especially robust in comparison to linear layerwise theory due to the number of discrete layers used to model the thick laminate through thickness and in the prediction of interlmainar shear stresses at the interface. In addition, the theory used also offers a better range of applicability due to the better accuracy offered. Over the years, many papers investigating the effects of interlaminar stresses had been published. The finite difference method with classical elasticity theory was used by Pipes and Pagani for determining the behaviour of finite width laminate in uniform axial strain and where interlaminar stress at the free edge is found to be of a significantly huge amount. Other studies soon ensued such as the perturbation solution techniques by Hsu and Herakovich, the finite difference method using large elements with complex stress field by Rybicki; and Wang and Crossman finite difference method, as well as the approximate analytical solution by Pagano and Wang and Choi. However, all of these studies involve the interlaminar stresses at the free edges of finite composite laminates. It is however, well acknowledged that interlaminar stresses arise such as to satisfy equilibrium at locations with in-plane stress gradients (Saeger, Lagace Dong ,2002), and material discontinuity within a structure is another source of arising in plane stress gradients, and therefore, interlaminar stress appear near the material discontinuities. (Tahani, 2005) Rose/ Herakovich, in 1993, further explore the force balance method of Kassapoglou/Lagace with the introduction of additional terms for the consideration of the discontinuous change of the elastic material properties in the interfaces and which accounted for the local mismatches in Poisson ratio and coefficient of mutual influence between adjacent layers. There are reported improvements in the resultant stress field. However, such improvements are also accompanied with a more demanding computational effort for the minimization of the complementary potential. In a similar study done by Kim/Atluri in 1995, thermal and mechanical loads were analyzed by assumed stress shapes which also accounted for both the local mismatches in Poisson ratio (similar to that of Rose/ Herakvich) and coefficient of mutual influence by applying respective mismatch terms in the stress representations. An approach that agreed to equilibrium demands and the given boundary conditions, the unknown stress functio ns were determined by application of the principle of minimum complementary energy of the laminate. The principle of minimum complementary theory was used by Bhat and Lagace (1994) to evaluate the interlaminar stresses at material discontinuities. In their analytical model, the laminate is formed by the merging of two area