Composite Plates Under Concentrated Load on One Edge and Uniform Load on the Opposite Edge

Transcription

1 Mechanics of Advanced Materials and Structures, 7:96, Copyright Taylor & Francis Group, LLC ISSN: print / online DOI:.8/ Composite Plates Under Concentrated Load on One Edge and Uniform Load on the Opposite Edge Christos Kassapoglou and George Bauer Department of Aerospace Engineering, Delft University of Technology, Delft, Netherlands CAE Associates, Inc., Middleury, CT, USA Downloaded By: [Technische Universiteit Delft] At: 7: April An approach to determine the stresses in a composite plate under a concentrated load on one edge and uniform load on the opposite edge is presented. It uses an exact solution of the governing partial differential equation otained with the use of Fourier series. The solution is compared to finite element results and is found to e in excellent agreement. The solution to this prolem can e used in the analysis of the effect of a stiffener termination in a panel or the transition from a stiffened panel to a sandwich or monolithic construction. Examples of dimensions and layup required to transfer the concentrated load into a composite plate are given. Keywords Polymer-matrix composites, plate theory, modeling, stress transfer. INTRODUCTION The design requirements of composite plates or shells in airframe structures often include transition of a stiffened design to a monolithic or sandwich design. The most common example is that of a stiffener termination [ ]. The local stiffness mismatch and eccentricities due to the termination, lead to localized ending and in-plane loads that may result in premature failure of stiffened panels. In other cases [], grids in grid-stiffened panels flare out into the skin of the panel to facilitate attaching to adjacent panels. Transitioning the load from the axial memer into a non-stiffened plate causes a local stress concentration which must e properly accounted for during design to avoid failure. In all these cases, oth memrane and ending loads act on the plate eyond the stiffener termination and a complete analysis would have to account for oth. As a first step, an analysis to determine the memrane loads is presented here. The situation is shown in Figure. A concentrated load is exerted on one edge of a composite plate. The width over which the load is distriuted is small compared to the width of the plate Received July 8; accepted April 9. Address correspondence to Christos Kassapoglou, Dept. of Aerospace Engineering, Delft University of Technology, Delft, Netherlands. anniekar@ath.forthnet.gr and can represent the flange width of a stiffener that terminates at the edge of the plate. The axial load in the stiffener is transferred into the plate. The concentrated load is reacted at the other end of the plate y a uniform load. Note that the simplified prolem at the ottom of Figure will not model the stiffened panel transferring load into a flat panel accurately if the stiffeners are so close that the stress field in the flat plate caused y one stiffener interacts with that of adjacent stiffeners. This limitation will e further investigated later.. CLOSED FORM SOLUTION A discussion of the prolem of Figure for isotropic plates can e found in Timoshenko and Goodier [5]. There, Fourier series are used to solve a case where a concentrated load acts on oth ends of an isotropic plate and the case of a uniform distriuted load on one end is otained at the center of the plate in the limiting case where the plate is very long. Here, an analogous approach is used ut with the added complication of material anisotropy. With reference to Figure, and asic mechanics of anisotropic plates, see for example [6, 7], the stress-strain equations Nx = A ε x + A ε y Ny = A ε x + A ε y Nxy = A 66 γ xy averaged over the thickness H of the plate ecome σ x = A H ε x + A H ε y σ y = A H ε x + A H ε y () τ xy = A 66 H γ xy where A ij are plate memrane stiffnesses and the plate is assumed to have no stretching-ending coupling (symmetric layup) and no stretching-shearing coupling (A 6 = A 6 = ). 96

2 COMPOSITE PLATES UNDER CONCENTRATED AND UNIFORM LOAD 97 F Stiffened panel one otains y successive differentiation: τ xy x y = σ x x (6) Transitioning into flat panel σ y y = τ xy x y (7) Eqs (6) and (7) can e comined to yield, σ y y = σ x x (8) Equations (6) and (8) can e placed into Eq. () to give: Downloaded By: [Technische Universiteit Delft] At: 7: April Simplified prolem to e solved FIG.. x σ h a σ o Prolem under consideration. y Plate thickness, H Solving the first two of Eq. () with respect to ε x and ε y gives: ε x = HA σ x HA σ y A A A ε y = HA σ y HA σ x A A A Eliminating the displacements in the strain-displacement equations, one otains the strain compatiility relation: () γ xy x y = ε x y + ε y () x Placing Eqs. () and the last of Eq. () into Eq. () gives, A A A A 66 From the equilirium equations τ xy x y = A σ x y A σ y y + A σ y x A σ x x () σ x x + τ xy y = τ xy x + σ y y = (5) [ A A A ] σ x A A 66 x + A σ x y + A σ y x = (9) Differentiating Eq. (9) twice with respect to y, [ A A A ] σ x A A 66 x y +A σ x y +A σ y x y = () Also, differentiating Eq. (8) twice with respect to x, σ y x y = σ x x () and sustituting into Eq. (), the final equation for σ x is otained: [ σ x x + A A A A ] σ x A A 66 A x y + A σ x A y = () This equation is to e solved suject to the oundary conditions: σ x (x = ) = for y h and + h y () σ x (x = ) = σ = F h for y + h Hh σ x (x = a) = σ o = F H () σ y (y = ) = σ y (y = ) = (5) τ xy (x = ) = τ xy (x = a) = τ xy (y = ) = τ xy (y = ) = (6) F is the total force acting on each edge of the plate and h is the width over which F is distriuted on edge x = of the plate. It is straightforward to verify that Eq. () accepts solutions of the form σ x f n (x) cos nπy (7)

3 98 C. KASSAPOGLOU AND G. BAUER Downloaded By: [Technische Universiteit Delft] At: 7: April Sustituting in Eq. (), f n (x) is found to satisfy the following equation, where d f n dx ( nπ β ) d f n dx ( ) nπ + γ f n = (8) β = A A A A A 66 A A γ = A A (9) The solution to Eq. (8) is in the form f n = De φx with φ =± ( nπ ) β ± β γ () If the plate were infinitely long the two solutions in Eq. () with positive real parts would e neglected to avoid infinite stresses. It will e assumed here that the plate is long enough so that the solution for an infinite plate is still valid. As will e shown later, for most practical cases this is sufficient. For very short plates, all four values of ϕ should e used. Using the solution for f n to sustitute in Eq. (7) and noting that Eq. () also accepts constants as solutions, the longitudinal stress σ x in the plate can e written in its most general form as: σ x = Ko + n= A n [ e φ x + C n e φ x ] cos nπy () where Ko, A n, and C n are unknowns. Using Eq. (), the equilirium Eqs. (), and the oundary conditions (5) and (6), it is found that the index n must e 9 even and the stresses are given y (sustituting n for n), σ x = Ko + σ y = n= τ xy = n= ( nπ n= A n ( e φ x φ ) e φ x φ cos nπy ) ( φ A n φ e φx φ e ) ( φ x cos nπy nπ φ ( A n e φ x e ) φ x sin nπy ) () The two remaining unknowns Ko and A n are determined y satisfying the remaining oundary conditions () and (). From the first of Eqs. (), σ x can e treated as a Fourier cosine series. Then, Ko must equal the average of σ x at any x location, Ko = F H () Then, at x = a, the exponentials in the σ x expression are zero (since the plate is assumed sufficiently long) and σ x = Ko there, which satisfies Eq. (). Equation () can now e satisfied y determining A n as coefficients of Fourier cosine series. This is done y placing the first of Eq. () into Eq. () multiplying oth sides y cos(qπy/), with q an integer, and integrating with respect to y from to. The result is, A n = F φ nπh cos nπ sin hh φ φ nπ () In practice, the series in Eqs. () are truncated. The numer of terms to e used is selected y comparing the distriution of σ x at x = with the exact distriution that reproduces the concentrated load at x =. This is done in Figure for the case of a (±5) s layup with properties given in Tale. Stress is normalized with the uniform stress at x = a. For the purposes Stress/Far field stress ,,,,,5,6,7,8,9 Normalized distance y/ n= n= n= n=8 n=6 exact FIG.. Stress at the panel edge where concentrated load is applied as a function of numer of terms in the series.

4 COMPOSITE PLATES UNDER CONCENTRATED AND UNIFORM LOAD 99 Downloaded By: [Technische Universiteit Delft] At: 7: April Property TABLE Material properties used Value E 7. GPa (. msi) E 7. GPa (. msi) G 5.7 GPa (.75 msi) ν.5 Ply thickness.95 mm (.75 in) of comparison, the dimensions of the panel are a = 5.8 cm ( in), = 5. cm (6 in) and the concentrated load is over a distance h =.778 cm (.7 in). The applied force F is 89 N ( l). As can e seen from Figure, the exact shape of the stress distriution requires a large numer of terms. Even for n = 6 there is a small difference etween the exact distriution and the prediction. On the other hand, the difference etween n = 8 and n = 6 is small enough that it was decided to use n = 8 in all cases from now on. As will e shown later with comparison to a finite element solution, this provides sufficient accuracy in all cases examined.. FINITE ELEMENT ANALYSIS PREDICTION To validate the closed form solution for stress distriution within the composite plate, a fine mesh finite element model was created. Panel dimensions of 5. cm (6 inches) in width and 5.8 cm ( inches) in length were used. (The 5.8 cm length was aligned with the X axis of the gloal cartesian coordinate system). ANSYS v. was used to generate a fine mesh with the focus of using the linear static analysis solution. First order isoparametric shell elements were used. Element edge lengths of.5 to.75 in. were used to transition a refined mesh near the load application to coarse density near the far-field constraint oundary condition. This generated 7,8 SHELL8 elements and,69 degrees of freedom. The model and applied loading were configured as shown in Figure. The finite element model was generated with a very fine mesh in the load application area in order to accurately capture the stresses in the high gradient region. Such a fine mesh was necessary for more accurate comparison to the analytical solution. Susequently, matching the analytical method presented here to the finite element results would increase the confidence in the analytical model. The element properties were the same layup, (±5)/(/9)/ (±5), with ply material properties as defined in Tale. Element material coordinate directions were defined to e aligned with the gloal cartesian X-Y system. ANSYS shell section definition was used to define the ply layup stacking sequence with respect to the element material coordinate system. An applied load of 5 N (, l) was used acting in-plane over a discrete.78 cm (.7 in) edge length representing the load introduction from a typical stringer flange. The displacement oundary conditions were set such that the panel reacts to the FIG.. Fine mesh finite element model, applied loads, and oundary conditions.

5 C. KASSAPOGLOU AND G. BAUER Downloaded By: [Technische Universiteit Delft] At: 7: April FIG.. applied load uniformly in the X direction and is ale to expand naturally due to Poisson s effect in the Y direction. The ANSYS linear static solution was performed. The results and oundary conditions were checked to verify the model reproduced a σ stress on the applied load edge and confirmed proper level of uniform far-field stress sufficiently far from the oundary conditions. Checks also were performed to confirm far-field total load reaction. The figures presented here (Figures 6) are contour plots demonstrating stress distriution within the composite panel. It is interesting to oserve, in the figures, the panel deflection as well as the localized dissipation of stresses directly eyond the load introduction. Detailed numerical comparison of the finite element results with the proposed analytical solution is presented in the following section. Additional finite element model verification was performed to assess the adequacy of the mesh density. Two additional Finite element model axial X-direction stress contour plot. coarse mesh models were run to converge on an element size that would produce the most confidence in the results. Tale compares the smallest element mesh size near the applied load area along with peak normal, transverse, and shear stress results. In each case, the stress values are normalized y the peak values of the previous mesh. The final column in Tale shows that the final mesh density has essentially converged on the est practical mesh for this study. It is interesting to note that the peak shear stress converges more slowly than the other two stresses in Tale.. CORRELATION WITH FINITE ELEMENT ANALYSIS The normalized axial stress along the center of the panel is compared to the finite element model prediction for the (±5)/(/9)/(±5) layup in Figure 7. The finite FIG. 5. Finite element model transverse Y-direction stress contour plot.

6 COMPOSITE PLATES UNDER CONCENTRATED AND UNIFORM LOAD Downloaded By: [Technische Universiteit Delft] At: 7: April FIG. 6. element results are in excellent agreement with the analytical solution. The comparison of the two prediction methods for transverse stress as a function of normalized transverse distance at the x =.8 cm (.5 in) location is shown in Figure 8. This particular location was chosen ased on the element size used in the finite element model. A path plot through the finite element model at this section was used to extract stress results for correlation. This provides several rows of elements away from the load introduction point to minimize the effects of point-load oundary conditions. Using element center results as opposed to grid point averaged results provides more accuracy as the results are calculated using shape functions from the element s exact integration points. The analytical and finite element methods are in good agreement. The analytical solution predicts a peak that is % higher than that of the finite element solution. Also, there are two short plateaus in the finite element solution on either side of the peak, that the analytical solution does not accurately predict. It is elieved that using more terms in the series of the analytical solution would improve the accuracy in this case. Finite element model shear stress contour plot FIG. 7. Axial stress/ Far field stress.59 Normalized longitudinal distance (x/a) Theory FEA Normalized axial stress σ x H/F along the center of the panel. TABLE Mesh density study Starting Improved Final mesh A mesh B mesh C Smallest element edge length (mm) Numer of elements Peak axial stress Peak transverse stress..6. Peak shear stress Transverse stress/far field stress Theory Normalized transverse distance y/ FIG. 8. Normalized transverse stress σ y H/F as a function of distance y evaluated at x/a =.75. FEA

7 C. KASSAPOGLOU AND G. BAUER Downloaded By: [Technische Universiteit Delft] At: 7: April Shear stress/far field stress Normalized transverse distance y/ Theory FIG. 9. Normalized shear stress τ xy H/F as a function of distance y evaluated at x/a =.75. The shear stress comparison is shown in Figure 9 as a function of transverse distance across the panel at the same x location, x =.8 cm (.5 in). The analytical method proposed here is in excellent agreement with the finite element results. 5. RESULTS The excellent agreement with the finite element results gives confidence in the present method, which can e used to otain the stress distriutions for different plate layups and geometries. This will give an idea of the size and layup of reinforcement necessary to transfer the load from a stiffened panel to a monolithic or sandwich panel. Typical results are presented elow for three different panel layups, (±5), (/9), and the quasi-isotropic [(±5)/(/9)]s. These layups are fairly typical of sandwich panels where each facesheet of the sandwich consists of two of the four plies specified. The core type and thickness do not affect the in-plane stress distriution. The panel length a is 5.8 cm ( in), the width is 5. cm (6 in) and the concentrated load of l is over a distance h =.778 cm (.7 in). The axial stress σ x along x at the center of the panel is shown in Figure. The peak stress is the same for all three layups since the layup thickness, applied load and the width over which it acts, are all the same. As x increases however, the rate at which the stress decreases depends strongly on the layup. The (±5) FEA Transverse stress/far field stress (±5) [(±5)(/9)]s (/9) Normalized transverse distance y/ FIG.. Normalized transverse stress σ y H/F as a function of distance y evaluated at x/a =.5. has the steepest rate of decay, followed y the quasi-isotropic layup. The (/9) layup, which has no 5 degree plies, has the slowest rate of decay. At the far field, (x large) all three curves go to the uniform stress at the x = a edge of the panel. However, ecause of the different decay rates, the reinforcement for the (/9) layup, needed to accommodate the stresses in excess of the far-field stresses, should e significantly longer that the other two. Assuming that a reinforcement that covers the region over which the maximum stress drops to within % of the far field value is adequate, the length of the reinforcement would correspond to x/a =.5 for this layup while it would correspond to x/a<. for the other two cases. The transverse stress σ y is shown in Figure as a function of y at x =.5 mm (. in) from the x = edge. Here, the (±5) layup shows a peak that is significantly higher than the value of the other two layups. So while the axial stress σ x for the (±5) layup was shown to die down to its far-field value much faster than for the (/9) layup and thus require less of a transition region, the σ y stress is twice as high as for the (/9) layup suggesting that the (±5) layup may e more critical. The peak value of this stress should also e taken into account when the reinforcement is designed. Finally, the shear stress τ xy isshowninfigureasafunction of y at the same x location (x/a =.5). As in the case of transverse stress, the (±5) layup has higher stresses than the other two layups. Again, the peak stresses can e significant and should e accounted for in the design of the reinforcement Axial stress/far field stress (±5) [(±5)(/9)]s (/9) Normalized longitudinal distance (x/a) FIG.. Normalized axial stress σ x H/F as a function of distance from the edge at the center of the panel. Shear stress/far (±5) field axial stress [(±5)(/9)]s (/9) Normalized transverse distance y/ FIG.. Normalized shear stress τ xy H/F as a function of distance y evaluated at x/a =.5.

8 COMPOSITE PLATES UNDER CONCENTRATED AND UNIFORM LOAD Downloaded By: [Technische Universiteit Delft] At: 7: April Similar plots can e otained for the axial stress σ x versus y (not shown here for conciseness). The corresponding results at other x locations (also not shown), show similar distriutions ut the peaks decrease significantly as x increases. Therefore, a weight-optimum local reinforcement, should have thickness and layup tapering oth in the x and y directions and should e such that the comination of σ x, σ y, and τ xy does not lead to failure at any given location. The dimensions of the reinforcement can e otained from graphs like those in Figures through as the distances over which the stresses are significantly higher than the far-field stresses. For example, on the asis of the results in Figures and, the transverse and shear stresses are confined in a region.5 y/.65. This would suggest a douler width dy such that dy/y = (.65.5) =., which, for the present case, leads to a value dy =.57 cm (.8 in). This value also covers the σ x distriution for these three layups. It is important to note that this value of dy is significantly larger than the width h (=.778 cm or.7 in) over which the applied load is distriuted. Thus, in the case of a terminated stiffener, creating a local reinforcement with width equal to the stiffener flange width may not e sufficient. In addition, the rate of decay of σ x in Figure suggests a reinforcement length dx such that dx = (.5 a) for (/9) and.a for (±5) or [(±5)/(/9)]s layups (giving dx = 5. cm or in and dx = 5. or 6 in respectively). These should e viewed as guiding examples only. The exact dimensions and layup of the reinforcement should e decided on a case-y-case asis using the analysis presented earlier as a starting point. Once a configuration has een decided upon, detailed analysis of the plate with the local reinforcement in place is necessary. One final comment aout the effect of plate length or stiffener spacing is in order. As long as the plate is long enough so the stresses in Figure have decayed close to their far-field values efore reaching the other end, the solution presented is valid. If the plate is shorter than that, (shorter than.5a for (/9) )the solution must e modified and four exponentials using all values of φ in Eq. () should e used. Also, if the stiffener spacing is greater than the range dy determined ased on Figures 8 and 9 (. ), the stress distriution caused y one stiffener is not affected y the adjacent stiffeners and the solution is valid. If this is not the case, the solution should e modified to account for multiple load introduction points. The approach is the same as efore and only the expression of the coefficients A n changes as the oundary condition () at x = must reflect the presence of more than one stiffeners introducing load into the panel. 6. SUMMARY AND CONCLUSIONS An approach to determine the in-plane stresses in an anisotropic plate loaded y a concentrated load at one end and uniform load distriuted over the whole edge at the opposite end was presented. The method gives the stresses in closed form and was found to e in excellent agreement with very detailed finite element results. One of the advantages of the method is that it can e used to otain preliminary design (dimensions and layup) of the reinforcements necessary to transfer the concentrated load into the plate without failure. This is preliminary as the presence of the reinforcement will alter the load locally and the solution must e iterated upon to otain the final design for the reinforcement. It was found that the axial stresses in the panel decay more slowly for (/9) panels than for (±5) or quasi-isotropic panels. On the other hand, the peak transverse and shear stresses in the (±5) panels are significantly higher than in the quasi-isotropic or (/9) panels. These peaks must e accounted for in the reinforcement design to avoid failure. REFERENCES. D.C. Jegley, Behavior of Compression-Loaded Composite Panels with Stringer Terminations and Impact Damage, Proc. 9th AIAA/ASM/ASCE/AHS 9th Structures, Structural Dynamics and Materials Conference, Long Beach CA, 998, AIAA Paper pp , C.H. Shah, H.P. Kan, and M. Mahler, Certification Methodology for Stiffener Terminations, Final Report, FAA report No DOT/FAA/AR-95/, M.J. Shuart, D.R. Amur, D.D. Davis, R.C. Davis, G.L. Farley, C.G. Lotts, and J.T. Wang, Technology Integration Box Beam Failure Study: Status Report, Proceedings of Second NASA Advance Composites Technology Conference, Lake Tahoe, NV, (NASA Conference Pulication 5), pp. 99, 99.. D.J. Baker, D.R. Amur, J. Fudge, and C. Kassapoglou, Optimal Design and Damage Tolerance Verification of an Isogrid Structure for Helicopter Application, Proceedings of th AIAA/AHS/ASME/ASCE/ASC Structures, Structural Dynamics and Materials Conference, AIAA Paper AIAA, pp. 5,. 5. S.P. Timoshenko, and J.N. Goodier, Theory of Elasticity, rd Edition, McGraw-Hill Book Co., R.M. Jones, Mechanics of Composite Materials, nd Edition, Taylor & Francis, C.Y. Chia, Nonlinear Analysis of Plates, New York, McGraw Hill, 98.