Advances in the Bonded Composite Repair o f Metallic Aircraft Structure phần 7 pdf

Chapter 1 I. Thermal stress analysis 323 derive an expression for E, in terms of these quantities. From equilibrium considerations we have: hence: where: (11.22) (1 1.23) The expression for the stress state in the plate just outside the patch is given by Eq. (11.16) for r = RI: c3 1 c2 o=El [(I -v) (1 +v)R; (1 1.24) We will now derive the expressions for the stress state in the plate beneath the patch and in the patch. From Eq. (1 1.15) with r 5 RI, under a uniform temperature, the displacement is given by: Since the displacement is the same in both the patch and plate we have: (1 + V)QTI + - U +c2=- 2 I-RI ’ RI ’ ( 1 + v)a2 TI U where the displacement u corresponds to the location r = RI. The radial stresses for the plate and patch are given by: (11.25) (11.26) (11.27) (1 1.28) (11.29) Using Eqs. (1 1.26-1 1.29) we have the expressions for the radial stresses in the plate 324 Advances in the bonded composite repair of metallic aircraft structure beneath the patch and in the patch: (1 1.30) (11.31) To obtain the residual stress in the plate beneath the patch it is necessary to sum Eqs. (11.13) and (11.30), but with TI = -TI in Eq. (11.30). Hence the final expression for the residual stress beneath the patch is: ( 1 1.32) Since the initial stress in the patch is zero, then the residual stress in the patch is given by Eq. (11.31), but again with TI = -TI hence: (1 1.33) and the final expression for the residual stress just outside the patch is given by the summation of Eqs. (1 1.13) and (1 1.24), hence: ( 1 1.34) These equations now give the residual stress in terms of the cure temperature T. The displacement u at r = RI for Eqs. (1 1.17, 11.18) and integration constants C2, C3 are given in the appendix. These equations now give the residual stress in terms of the cure temperature T. For operating temperatures different from room temperature, Eqs. (1 1.24), (1 1.30) and (1 1.31) can be used to calculate the stresses. In this case TI= TO = uniform temperature change from room to operating temperature. The final stresses are obtained by superimposing these stresses on the residual stresses. 11.2.1. Comparison of F.E. and analytic results The solution of these equations has been carried out for AT = lOO"C,and the following quantities have been evaluated for the comparison with F.E. results, the mesh is shown in Figure 11.4: 1. residual stress just outside the patch at r = RI Chapter 11. Thermal stress analysis 325 Fig. 11.4. Finite element mesh of circular patch on circular plate. Here RI = 162 mm and Ro = 500 mm. 2. residual stress in the skin beneath the patch (01) 3. residual stress in the patch (02) As an example a circular patch and plate are considered whose mechanical properties are shown in Table 1 1.1. These properties are representative of a quasi- isotropic boron patch reinforcement of an aluminium plate (although the value of a used here for boron corresponds to uni-directional boron and should have been Table 11.1 Material properties used for study of circular repairs on circular plates, AT = 100°C. Young's Coefficient of Thickness modulus Poisson's thermalexpansion Conductivity Component (mm) (MPa) ratio (1°C) (J/ms"C) Plate(A1uminium) 1 .O 71016 0.3 23 x IOs6 13.2 Patch(boron) 0.5 156000 0.3 **4.1 x 10-6 0.294 *Adhesive (FM300) 0.254 3460 0.35 * Only used for a 3D run. In this instance the value for the laminate is taken to be equal to the unidirectional value. ** 326 Advances in the bonded composite repair of metallic aircraft structure 3.76 x 10@ for the laminate). While this is not a perfect representation of an actual repair it is acceptable for estimating residual stresses. Also this assumes that bending is restrained, e.g. by stiffeners or thick plates, and also the edge restraint exists when the repair is bounded by structural elements such as spars and ribs. Consider the case in which the plate edge is restrained in the radial direction. Firstly consider the heating up process to the cure temperature, given by Eq. (11.13) and shown in Figure 11.5. The comparison between theory and F.E. results is in agreement to four significant figures. In the case of no edge restraint the initial stresses would be zero. The second stage of the process involves cooling down from the cooling temperature alone. The analytical and F.E. results are shown in Figure 1 1.6 where the curves are from Eqs. (11.24), (11.30) and (11.31), and the points on the curves are F.E. results. In all cases very good agreement between analytical and F.E. results are obtained, (to four significant figures). The final solution for the residual stresses is given by Eqs. (1 1.32-1 1.34) in Figure 11.7 with the corresponding F.E. results. Again very good agreement between analytical and F.E. results is obtained. Note that the residual stresses in the plate shown in Figure 11.7 are significantly lower than those during the cooling process, Figure 1 1.6. This is simply due to the lack of initial stresses which arise as a result of the restrained edges of the plate when heated up to the cure temperature. The assumption of edge restraints is important. Typically a repair to an aircraft wing plate can be considered as fully restrained in the radial direction if the repair is bounded by significant structural elements such as spars and ribs. Returning to the residual stresses shown in Figure 11.7. It is evident that for large values of Ro/R, asymtotic values occur for all stress components and may be considered as limiting values. -100, -105- -110- 2 -115- A 5 cn -120- cn a, 3 -125- F.E. -145(. , . I I I., . , . , , I 0 5 10 15 20 25 30 35 RdRi Fig. 11.5. Initial stresses in plate due to heating to cure temperature. Chapter 1 1. Thermal stress analysis 0- -20- -40- -60- 327 rr n 0 0 Stress in patch, Theory/F.E. Stress in plate outside patch,Theory/F.E. 1 Stress in patch,Theory/F.E. 0 Stress in plate beneath patch,Theory/F.E. 0 Stress in plate just outside patch, Theory/F.E. 501 -504 0 5 10 15 20 25 30 35 R,IRi Fig. 1 1.6. Comparison of theory and F.E. results for cooling down process only. So far, the adhesive has not been considered in the analysis. However F.E. results have been obtained in which the patch and plate have been coupled using 3D adhesive elements. To make a useful comparison with the previous analytical work the bending of the plate has had to be restrained. The introduction of the adhesive has 328 Advances in the bonded composite repair of metallic aircraft structure Table 11.2 2D isotropic circular plate, patch and 3D adhesive elements. Residual displacements and stresses, al = 23 x 10-6/"C, a2 = 4.1 x 10-6/"C (analytic values in parenthesis), [5]. Residual stress Residual stress in Displacement at just outside patch skin beneath patch Residual stress Ro (mm) edge of patch (mm) (MPa) (MW in patch (MPa) 500 - 0.1 1551 128.31 161.91 - 65.532 (-0.1 1499) (127.92) (1 6 1.33) (-66.813) resulted in an error of only 2% in direct stresses, shown in Table 11.2, and indicates that the use of a closed form solution is sufficiently accurate for patch design. The main concern in this chapter is the equations for direct stresses in the repair. It is important to know the direct stress in the plate beneath the patch in order to predict crack growth rate or simply the residual strength of the repaired structure. The adhesive itself has no effect on the maximum value of the direct stresses. Some F.E. programs have the capability in which the material properties can be temperature/time dependent. In this case a simulation of the bonding process can be carried out. The adhesive properties change during the curing process. At the end of the curing process the adhesive has developed a shear stiffness and as the repair is cooled to room temperature residual stresses develop. If the simulation capability is available, then residual stresses are directly obtained from the analysis. If this capability is not available then a superposition procedure can be used. The analysis is carried out in two steps. The first analysis is equivalent to heating up of the plate to the curing temperature (without the patch, since the adhesive has no stiffness at this stage). Secondly, another analysis is carried out with the patch included, subject to a cooling temperature equal to the cure temperature. ( TI = -TI). In the work presented here this two stage procedure has been shown to be very accurate. The superposition of these two analyses gives the residual stresses in the repair. Since the adhesive shear modulus is temperature dependent, an arithmetic average value of the shear modulus should be used during the cooling process. I I .2.2. Orthotropic solution Recent work by [7,8] has extended the analysis of residual stresses to circular orthotropic patches on isotropic plates. The solution of this problem is based on an inclusion analogy, which refers to the inner region of the repair where r 5 RI in which equivalent properties of the inclusion can be made without altering the stress or displacement state. Exact solutions are presented for both residual stresses and thermal coefficients of expansion. Results of this work are shown in Figures 11.8 and 11.9 for stresses due to cooling only. For plate stresses beneath the patch both oXx and oYY are predicted, as shown in Figure 11.8 and correspond to clamped edge conditions. As a comparison, the isotropic solution is included and gives close results when Chapter 1 1. Thermal stress anaIysis 329 Q a .4 - c .e , , , I I clamped edge 1.4 1.2 1 .o 0.8 0.6 o FEresults Equivalent isotropic patch 13 5 7 9 I1 13 Ratio of outer radius to inner radius RJR, Fig. 11.8. A circular patch over a concentric plate with outer edge being clamped: cooling induced stresses in the plate. [7] 1 .o F 6 0 FEresults W 0 - w Equivalent isotropic patch g 0.5 E P 2 2 -0 .* c v) E s t b -0.5 1 3 5 7 9 I1 Ratio of outer radius to inner radius Mi Fig. 11.9. A circular patch over a concentric plate with outer edge being clamped: cooling induced stresses in the orthotropic reinforcement. [7] 330 Advances in the bonded composite repair of metallic aircraft structure compared with oxx stresses, however the isotropic solution predicts oxx = oyy where in fact oyy < oxx. In the case of the reinforcement shown in Figure 11.9, the isotropic stress is slightly higher than the owx stress, but while the isotropic solution predicts oxx = oyy, where in fact the absolute value of the stresses, Joyy( < (oxx(. Note that for Ro/RI 2 1.5 the sign of the oyy stress is opposite in sign to the o, stress. For the cases of both the plate and reinforcement stresses, the isotropic solution is conservative. 1 I .2.3. Thermal stresses in a one-dimensional strip 11.2.3.1. Shear stresses. In a bonded joint in which two materials are bonded together, thermal stresses may develop as a result of the difference in thermal coefficients of expansion. The following equations will be given for a simple double overlap joint whose geometry is given in Figure 11.10 together with the location of the origin of the x axis. It is assumed that all the load transferred by the lap joint is by adhesive shear. Also, in this symmetrical joint it is assumed that no bending takes place [5]. The constitutive equations are: where: u1 and u2 are the displacements in the components of the joint a1 and a2 are the thermal coefficients of expansion E1 and E2 are the Young’s moduli of the two materials AT is the change of temperature = TC - TA Tc is the cure temperature TA is the initial temperature 61 and 02 are the stresses in the components of the joint. The equilibrium equations are: where: r1, t2 are the different thicknesses of the two components and z is the shear stress in the adhesive. Compatibility requires that the shear stress is given by: Ga ta z=-(u2-u,), (11.35) (1 1.36) (1 1.37) where: t, is the thickness of the adhesive Chapter 1 1. Thermal stress analysis 33 1 Reinforcement A. -x 2tl /! I Adhesive /w h t 1 t 1 OZ +- i t L Deformation of adhesive Fig. 11.10. One dimensional equation, definition of parameters. G, is the shear modulus of the adhesive. From these equations the result is obtained: (1 1.38) 332 Advances in the bonded composite repair of metallic aircrafi structure where: For thermal stresses only crA = 0 and: T = -(E1 Gaz - a2)ATe-X/' fa The direct stress for component 1 given by: (1 1.39) (1 1.40) (1 1.41) At x = 0, ul = 0, and 01 rises to a maximum value when x is large which is given by: (1 1.42) If we consider a single lap joint in which bending is restrained and if the thickness of the skin is ts, then tl = t,. I I .2.4. Peel stresses Comparisons of the F.E. are made with the d.e. expression, given by [ll]. This expression can be used for a single lap joint but with no bending and is a function of the shear stress T: up = -woEc/ta > (1 1.43) where (1 1.44) where x = (Ec/(4Dta))0'25 A = (z(ta/2~) sinxZ/2)/x3 E, =effective tensile modulus of the adhesive z = shear stress (normally taken to be the plastic value zp) D = E0tz/12(1 - v2) and is the bending stiffness of boron In this case the location of the coordinate system is at the midpoint of the joint, B = +(ta/2q COS X1/2)/X3 and x = k 112 corresponds to the ends of the joint. [...]... maximum value of 6 MPa, Figure 11.15 11.4 Application of analysis to large repairs of aircraft wings The object of this section is to determine the applicability of a closed form solution to the large repair of an aircraft wing, whose cross section is shown in Figure 11.16 In this case the repair covers one spar and two of the cells While the application of closed form solutions to repairs bounded by spars... provide the longitudinal (al) and transverse (az) coefficient of thermal expansion for a unidirectional laminate In this case the longitudinal coefficient of thermal expansion is a measure of the fibre property while the transverse coefficient is a measure of the resin property The starting point for the derivation of overall coefficients of thermal expansion in a laminate is the well known stiffness matrices... between the 2D F. E and 1D d.e do not necessarily indicate the existence of an error in either solution Factors that may influence results are firstly the Poisson's ratio effect which is not considered in the 1D d.e Secondly, in the F. E analysis shear deformation occurs in all components of the joint, while in the 1D d.e it only occurs in the adhesive Furthermore, the F. E analysis of bonded repairs in general... unidirectional laminate The resulting shape of the curve is dependent on both Poisson’s ratio effects (that are continuously changing), and plies values of a1 The coefficients of thermal expansion, a and a2, are the effective coefficients of 1 thermal expansion for a unidirectional laminate For a 3 D laminate the out of plane coefficient of thermal expansion may be taken as being equal to the value a2 in the unidirectional... isotropic solution and the second the orthotropic solution Application of a closed form solution to the repair of an aircraft wing requires an assumption of the value of Ro to be used Previous results have shown limiting stress values are obtained for large Ro/R, ratios, hence these are the results shown in Tables 11.9 and 11.10 Consider now the residual stresses predicted by the 2D isotropic solution From... restrained at the root end only Overall the 1D solution compares more favourably with the F. E than 2D sohtions, since with the wing tip unrestrained the problem is closer to a 1D problem The main reason for the difference of 2D and FE solutions is that the analytical models do not take into account the stiffness and temperature distributions of the spar webs and upper skin structure It is clear that for... adjusted for a cure cycle of 100 "Cand a room temperature of 25 "C and will be applied to the FE wing model The alternative procedure is to specify a convection heat transfer coefficient, h, in the thermal analysis, in which the rate of Fig 11.16 Cross-section of multi-spar wing (up-side down), flanges not shown 342 Advances in the bonded composite repair o metallic aircraft structure f 90P 80- 9 9? c 2 70 -... structural analysis then determines the distribution of initial stresses throughout the F. E model as shown in Figure 11.20 These stresses are spanwise components Note that tensile stresseson the edge of the wing box exist to restore the equilibrium .The next step in the analysis involves the cooling down of the F. E model which in this case also contains the boron patch In this case the sign of the temperatures... results are the same In the case just considered the F. E analysis was confined to the analysis of isotropic materials Now consider the case in which component 2 has orthotropic properties Composite materials are orthotropic and bonded joints are often comprised of such materials In the repair of cracked metallic structures, materials such as boron/epoxy laminates are used as reinforcement Consider a... in boron patch and stresses in plate surrounding the repair corresponding to a cooling thermal loading for a repair with a lay-up of [02, +45,03], Table 11.9 Stresses predicted in wing box corresponding to a geometric symmetric repair; layup [02, f4 5 ,O& boron repair AT = 75 °C Centre of plate under repair (MPa) Thermal loading Heating Cooling Residual Centre of patch (MPa) Plate just in/ outboard of . stiffness of boron In this case the location of the coordinate system is at the midpoint of the joint, B = +(ta/2q COS X1/2)/X3 and x = k 112 corresponds to the ends of the joint measure of the fibre property while the transverse coefficient is a measure of the resin property. The starting point for the derivation of overall coefficients of thermal expansion in a laminate. of a1. The coefficients of thermal expansion, a1 and a2, are the effective coefficients of thermal expansion for a unidirectional laminate. For a 3D laminate the out of plane coefficient