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Postbuckling analysis with progressive damage modeling in tailored laminated plates and shells with a cutout
Composite Structures 59 (2003) 199–216 www.elsevier.com/locate/compstruct
Postbuckling analysis with progressive damage modeling in tailored laminated plates and shells with a cutout De Xie, Sherrill B. Biggers Jr.
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Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA
Abstract An approach to modeling inplane damage progression in postbuckled laminated composite panels is shown to be accurate by comparison to experimental test data from other sources. A simple tailoring concept is shown to be very effective in increasing compressive buckling loads and ultimate loads for flat plates and curved panels with a central cutout. Effects of cutout size, the degree of tailoring, and inplane restraint on the unloaded edges are investigated. Optimal tailoring produces relative improvements in the flat plates ranging from 40% to 175% in buckling load and 190–240% in ultimate load capacity when compared to uniform plates with the same cutout sizes. In the curved panels, tailoring lowers the imperfection sensitivity and in some cases produces ultimate loads greater than the theoretical undamaged buckling loads. To the contrary, the ultimate load for the uniform curved panel is much lower than the undamaged buckling load. Relative improvements in ultimate loads range from a low of about 40% to a high of about 155% compared to uniform curved panels. Large differences in the damage initiation locations and damage progression patterns are shown between the flat and the curved panels. In summary, the tailoring concept investigated here can provide excellent improvements in ultimate load capacity in flat and curved panels with the largest benefits occurring in thin flat panels that are loaded far into the compressive postbuckling regime. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Damage progression; Postbuckling; Laminated plates and shells; Tailored design; Finite element analysis
1. Introduction Composite materials offer a natural opportunity to enhance structural integrity through the use of various laminate tailoring concepts. Among these is a simple stiffness tailoring concept which simply removes all axially oriented 0° materials from the central region of a rectangular panel and relocates the material in the edge regions of the panel. This approach has been shown to improve buckling loads, initial failure loads, and residual strength in laminated panels without affecting their weight in any way. The uniaxial stiffness tailoring concept described above was studied as a way to increase the buckling resistance of rectangular composite plates under uniaxial compression and increases of 200% or more were shown to be possible when compared to uniform plates [1]. A similar tailoring concept was investigated for shearing loads and buckling load improvements of up to
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Corresponding author. Fax: +1-864-656-4435. E-mail address: [email protected] (S.B. Biggers Jr.).
86% were achieved [2]. Furthermore, this approach was investigated for improved buckling resistance of rectangular composite plates subject to general in-plane combined axial, transverse and shear loadings [3]. Very large increases in buckling loads were found due to tailoring when transverse tensile loads are present along with axial compression. In many practical cases, panels are allowed to enter the postbuckling regime and operate at load and strain levels exceeding critical buckling values. Due to the higher load levels and the more complex structural response in the postbuckling regime, damage may initiate and propagate to some extent in panels and finally make them fail due to a combination of effects. Therefore a damage modeling scheme is a necessary component in the postbuckling analysis procedure. The previously mentioned research on initial buckling was extended to postbuckling with first-ply failure identified by the Tsai– Hill criterion [4]. This study showed that the compressive postbuckling stiffness and first-ply failure loads could also be greatly improved by tailoring. This work was further extended to consider the response after firstply failure in which buckling initiation, postbuckling
0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 2 3 3 - 7
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damage growth and ultimate failure were considered for rectangular plates in compression [5]. Damage growth in tailored flat plates and cylindrical shells containing a central cutout of varying size has also been investigated [6]. To simplify the structural response, this last work restricted the deformation to occur within the plane of the panels. In the case of tensile loading, the tailored region was found to arrest a temporarily unstable progression of damage. The current study releases the restraints on the outof-plane deformation in Ref. [6] and is a natural extension of the work in Ref. [5]. The damage is still assumed to occur within the plane of panels but now bending and twisting associated with the out-of-plane response also contribute to the damage development. No delamination or transverse damage mode is included. The analysis method was validated by comparison to experimental data from tests conducted by others. Results are presented for a set of flat and curved plates having a range of cutout sizes. In each case, different degrees of tailoring were also investigated so that the interaction with panel curvature, cutout size, and extent of tailoring could be evaluated. As expected, the flat plates were found to behave quite differently from the curved panels. This is contrary to the results presented in Ref. [6] where out-of-plane deformations were artificially restrained.
2. Tailoring approach A typical panel evaluated in this study is shown in Fig. 1. In the limit as the included angle approaches zero, the panel becomes a flat plate. The curved panel geometry considered in this study is a cylindrical shell with an included angle h ¼ 90°. Compressive loading was taken to be uniaxial in the x-direction and applied as a uniform end displacement. The panel was clamped on the loaded end and was simply supported on the
Fig. 1. Geometry of uniform and tailored panels.
Table 1 Mechanical properties of IM7/8551-7 and T300/5208 E11 (Msi) E22 (Msi) G12 (Msi) m12 XT (Ksi) XC (Ksi) YT (Ksi) YC (Ksi) S (Ksi)
IM7/8551-7
T300/5208
18.9 1.36 0.653 0.33 366.4 132.9 10.7 10.7 17.4
19.0 1.89 0.93 0.38 200.0 165.0 11.74 27.41 10.0
unloaded edges to prevent the panel from buckling as a wide column. Circumferential displacements on the unloaded edges were assumed to be either free or completely restrained, corresponding to upper and lower bounds that might be encountered in practical applications. The tailoring concept investigated here is a simple repositioning of all axially oriented 0° material into regions of certain widths near the edges of the panel. The baseline panel is square and has a uniform quasiisotropic layup [45/0/90]S with mechanical properties typical of IM7/8551-7 prepreg material (Table 1). Properties are also given for T300/5208 prepreg used for comparison to existing experimental data. The edge regions in the tailored panels are of total width b1 while the overall panel width (arc length) is b. The 0° material is uniformly distributed in the edge regions. The width ratio, b ¼ b1 =b, defines the extent of the tailoring and when b ¼ 1:0, the panel is uniform. When b < 1:0, the total thickness of the 0° material in the edge regions is increased such that the total weight of material in the panel remains unchanged.
3. Analysis method The analysis was conducted using the ABAQUS finite element code [7]. Only in-plane damage was taken into account, i.e. no interlaminar damage such as delamination was considered. The damage evolution model used is a simple ply-discount method in which stress and stiffness are reduced locally and instantaneously to zero once failure occurs at a point in a ply. The nonlinear behavior of the inplane shear modulus suggested by Chang [8] was also incorporated. The effect of this was to account for the slight softening of the shear stiffness as the stresses increased. The UMAT routine within ABAQUS was used to account for nonlinearities in the material due to local failures and softening of the shear modulus. These changes due to damage were introduced at individual integrations points as they occurred during loading. The Hashin failure criteria, as modified by Chang [8], were used to indicate damage initiation. To
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initiate out-of-plane deformation, an out-of-plane imperfection equal 1% of the panel thickness was introduced uniformly along the edge of the cutout. To demonstrate the accuracy of the method, the results predicted by the present analysis method were compared with the experimental results provided by Starnes and Rouse [9] for the flat plate (H4) and by Knight and Starnes [10] for the cylindrical shell (CP8). The geometry and finite element meshes of the plate and the shell are shown in Fig. 2. Both the flat plate (H4) and the cylindrical panel (CP8) were fabricated from unidirectional T300/5208 prepreg. Table 1 gives the mechanical properties except that E11 ¼ 19:6 Msi was used for panel CP8 as given in Ref. [10]. The loading and boundary conditions used for the analysis were selected to be consistent with the experimental tests. For the flat plate H4, the ply thickness is 0.00574 in. and the laminate stacking sequence is [45/0/90]3S . Engelstad et al. [11] also performed a similar progressive damage analysis on the plate. Fig. 3 compares the end reaction vs. end shortening curves obtained from present analysis with that from the experiment, and shows good correlation between the two. The plate supports increasing load after buckling until sufficient damage occurs to make it fail completely. The ultimate load predicted is within 4% of the ultimate test load and the corresponding strain is within about 7% of the test strain. The mode shapes at initial buckling (A) and ultimate load (B) shown in Fig. 4 agree with those given by Engelstad et al. [11] The same analysis procedure was applied to the CP8 cylindrical shell that has a ply thickness of 0.0056 in. and a laminate stacking sequence of [45/90/02 /90/45]S .
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Fig. 3. Comparison of end reaction vs. end shortening for H4, experimental [9] and analytical.
Fig. 4. Mode shapes for H4.
Fig. 2. Geometry and finite element meshes of plate H4 and shell CP8.
The predicted end reaction vs. end shortening curve is compared with the experimental data in Fig. 5. The analysis without considering damage shows a sharp drop in load after buckling, consistent with shell buckling. The damage analysis predicts damage development prior to, and an ultimate load slightly lower than, the theoretical buckling point. The damage analysis and experimental test results are remarkably similar. This is despite the fact that some delamination was observed at the free edge of the cutout in the physical test but was not taken into account in present analysis. The only major difference between the damage analysis and the test data is the much lower postfailure residual strength in the test, probably due to significant edge delamination after ultimate load. Fig. 6 shows the mode shapes at ultimate load (A) and a postfailure load (B). These deformation patterns agree well with those obtained by Knight and Starnes [10].
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Fig. 5. Comparison of end reaction vs. end shortening for CP8, experimental [10] and analytical.
Fig. 7. Finite element mesh for a typical flat plate.
Fig. 6. Mode shapes for CP8.
These comparisons of the analytical results to two rather different experimental test cases give sufficient confidence in the damage analysis to apply it to other cases in determining buckling and ultimate loads in the presence of inplane damage. This is done below, first for tailored flat plates with a central cutout and then for cylindrical shells.
4. Results for flat plates Now the simple tailoring concept described previously will be investigated for its effectiveness in increasing buckling and ultimate loads and also in controlling damage growth in flat plates with a central cutout. A typical finite element model for a uniform plate is shown in Fig. 7. Models for the tailored plates were similar. All plates are 6 in. by 6 in. and the uniform end displacement U is applied equally at each end of the
Fig. 8. Average running load Nx vs. end displacement U of flat plates: uniform and two extreme tailoring cases.
plate. Fig. 8 shows the average running load vs. applied end displacement curves for the uniform plate and two tailoring cases: the most highly tailored (b ¼ 0:10) plate and the most gently tailored (b ¼ 0:64) plate. The cutout size is 2r=b ¼ 0:3. Similar results obtained for other cutout sizes and other tailoring parameters will be shown later. The panel responses neglecting damage are also shown in Fig. 8 with dashed lines. The filled circles on the curves indicate load points for which deformed shapes will be shown in later figures. The Newton– Raphson method was used to perform this nonlinear analysis without any difficulties in convergence. Some important features can be observed in Fig. 8. Regardless of the specific tailoring, a well-defined buckling point is observed followed by a postbuckling
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response with reduced stiffness. The much higher postbuckling stiffness in the highly tailored plate follows the same trend found in earlier studies on plates without the central cutout [5]. The same can be said for the much higher ultimate load and corresponding overall panel strain. However for all three cases, the nearly complete coincidence of the curves with and without damage up until ultimate load suggests a sudden failure with no significant gradual damage progression prior to final failure. This is quite different from the tension load case considered in Ref. [6] where it was found that tailoring could arrest damage growth to a certain extent depending on the tailoring parameter. These observations can be better understood by examining the pictures of the deformed plates and the damage patterns presented below. 4.1. Three specific cases First, consider the uniform plate. Fig. 9 shows a sequence of deformed 1/4-plate models at load levels corresponding to the filled circles on the curve shown in Fig. 8. Deformed configurations are shown in an oblique view and scaled by a factor of 5 to exaggerate the deformation. Contours of out-of-plane displacement are cast on the deformed configurations using a fixed contour level with the same maximum and minimum value
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for all plots. The uniform flat plate was found (Fig. 8) to buckle at an end shortening U ¼ 0:0013 in. and this is reflected in the distinct geometry change around the cutout from U ¼ 0:0012 to 0.0013 in. The initial buckling mode shape is a single half wave over the entire plate. Fig. 9 also shows the development of the plate deformation during the postbuckling stage. The upward deformation around the cutout spreads mainly in the direction transverse to the load direction. The deflected shape becomes flatter near the center as the load and magnitude of the deformation increase, just as is the case in plates without a cutout. At an end shortening U approaching 0.0075 in., a downward deformation initiates away from the cutout, indicating a mode shift from one to three half waves along the load direction. The combination of these effects creates high bending and twisting gradients near the center of the unloaded edge. At the same time the large out-of-plane bending deformation around the cutout reduces the axial stiffness of the central region of the plate and makes this region shed compressive load toward the side regions. Therefore, the region near the center of the unloaded edges has the highest combination of inplane, bending, and twisting loads and damage initiates in this area. This combination of effects is in fact one of the reasons that the tailoring concept investigated here should prove to be effective.
Fig. 9. Out-of-plane displacements, uniform plate (b ¼ 1:00, 2r=b ¼ 0:3).
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Fig. 10. Damage, uniform plate just after ultimate load (b ¼ 1:00, 2r=b ¼ 0:3).
Fig. 10 presents plots showing both the matrix failure and the fiber failure in top four plies just after the ultimate load condition. The dark areas are the damaged areas. Based on the observations of the out-of-plane deformations discussed above, it is easy to see why the damage exists in the regions shown rather than at the sides of the cutouts as one might expect for purely inplane response. In general, the matrix failure of the þ45° ply is similar to fiber failure of the 45° ply and vice versa. (Note: the þ45° fibers run tangent to the side of the cutout in the quadrant of the plate shown here.) The same can be said for the 0° ply and the 90° ply. Although nonlinearity in the matrix shear modulus started devel-
oping at loads much less than ultimate, this did not have a significant effect on the overall response. It was observed that failures in the matrix and fiber started developing from a load very close to the ultimate and quickly propagated to the distribution shown in Fig. 10. This reinforces the idea of sudden failure mentioned previously. Similar deformation and damage developments were observed for the two tailored plates. Figs. 11 and 12 show the out-of-plane deformation configurations and the damage patterns, respectively, for the most gently tailored plate. In this case (b ¼ 0:64) the 0° material is broadly distributed from near the edge of the cutout to
Fig. 11. Out-of-plane deformation, most gently tailored plate (b ¼ 0:64, 2r=b ¼ 0:3).
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Fig. 12. Damage, most gently tailored plate just after ultimate load (b ¼ 0:64, 2r=b ¼ 0:3).
the unloaded edge. The dashed line marks the inner edge of the 0° material. This gentle tailoring greatly reduces the inplane stress concentration at the cutout and slightly increases the stiffnesses near the unloaded edge. However, since the cutout stress concentration did not prove to be crucial in the uniform plate, it is understandable that this gently tailored plate performs very much in the same way as the uniform plate. The buckling load is slightly increased as are the ultimate load and ultimate end displacement. But the out-of-plane displacements and the damage development patterns are nearly identical to those of the uniform plate.
Figs. 13 and 14 show deformations and damage patterns for the most highly tailored plate (b ¼ 0:10). Here the 0° material is concentrated in a narrow band near the unloaded edge. This localization of material may or may not be practical, depending on the application. The deformations appear quite similar to the other two cases except for the increased resistance to rotation near the unloaded edge. This tends to force the critical location and damage locations away from the unloaded edge to some degree. The main effect is to delay the initiation of fiber damage until a much higher applied end displacement and to retain a much higher
Fig. 13. Out-of-plane deformation, most highly tailored plate (b ¼ 0:10, 2r=b ¼ 0:3).
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Fig. 14. Damage, most highly tailored plate just after ultimate load (b ¼ 0:10, 2r=b ¼ 0:3).
postbuckling stiffness in the panel. These effects combine to give a greatly increased ultimate load. However, even this highly tailored design does not act to arrest damage growth and make the failure less sudden. Compared to the uniform plate, the highly tailored design is found to increase the postbuckling stiffness by 179%, the ultimate strength by 229%, and the ultimate axial strain by 39%. The corresponding improvements for the gently tailored design are 21%, 29%, and 5%. Furthermore, despite making the plate thinner in the center region, neither tailored design has higher outof-plane deformations than the uniform plate at a given end displacement. For the uniform plate at U ¼ 0:0125 in., the peak value for the upward and downward deformations are 0.137 and 0.0510 in., respectively. For the highly tailored plate, the corresponding values are 0.120 and 0.0547 in., while for the gently tailored plate the corresponding values are 0.124 and 0.0550 in. Therefore, at least for the cutout size considered above, both the most highly tailored and the most gently tailored designs offer improvement in every aspect of structural performance. Unfortunately, however, the desirable trait of damage arrest that was observed in Ref. [6] for tailored plates having a cutout and tensile loading does not present itself in the case of thin compressively loaded tailored plates. To give a more complete understanding of the effects highlighted above for three specific cases, now consider the response over a range of cutout sizes and a wider range of tailoring parameters. The average running load vs. applied end displacement curves are shown in Fig. 15 for the flat plates with four different cutout sizes. Results are shown for uniform ( b ¼ 1:0) plates and for tailored plates with b ranging from 0.10 to as high as 0.76 depending on the cutout size. Here it can be seen that the same trends discussed previously hold for the full range of parameters. Now the buckling loads and the ultimate loads obtained from this figure will be plotted vs. the tailoring width parameter and cutout size to further aid
our understanding of the effectiveness of tailoring on structural performance. 4.2. Initial buckling loads Fig. 16 shows the critical buckling loads (Ncr ) for each cutout size as a function of the tailoring width ratio ( b). Values for the uniform plates are also shown as separate points at b ¼ 1:0 for reference. These data are also shown as buckling load ratios (Rcr ) in Fig. 17 where the uniform plate critical buckling loads for each cutout size were used in the normalization. These plots show that tailoring can increase the critical buckling loads in a range from 24% to 175% with larger percentage improvements occurring with the larger cutouts and for intermediate values of the tailoring width ratio. As an example, with 2r=b ¼ 0:3, the optimum tailoring is with b ¼ 0:4 producing a 120% increase in buckling load (Rcr ¼ 2:20). Simulations were also performed for flat plates for which circumferential displacements on the unloaded edges were restrained. The effect of this restraint is to introduce a nonuniform transverse compressive load along the previously unloaded edges. Boundary conditions in most practical cases would be between the free and totally restrained cases considered here. Load vs. displacement plots very similar in character to those shown in Fig. 15 were obtained and they are not repeated here. However, some results are presented in Figs. 18 and 19, which summarize the buckling loads and corresponding Rcr values for the restrained edge cases. Compared to the free edge case, the following differences were found. First, the maximum value of critical buckling loads and Rcr values are lower for all cutout sizes. This is reasonable since the panel is effectively now loaded in biaxial compression. Second, the optimum design for buckling tends to shift from an intermediate degree of tailoring towards slightly gentler tailoring, i.e. the optimum b increases when the side
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Fig. 15. Average running load Nx vs. end displacement U of flat plates.
Fig. 18. Critical buckling loads, Ncr . Fig. 16. Critical buckling loads, Ncr .
Fig. 17. Critical buckling load ratios, Rcr .
Fig. 19. Critical buckling load ratios, Rcr .
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Next, consider the effect that tailoring has on the ultimate failure load. The peak values taken from each
load displacement curve in Fig. 15 are plotted against b in Fig. 22. For all cutout sizes, the maximum ultimate load is obtained for b ¼ 0:1. The same observation was reported for postbuckled plates without cutouts in Ref. [5]. The data in Fig. 22 were normalized by the ultimate loads of the uniform plate with the same size cutout and these ultimate load ratios ðRu Þ are plotted in Fig. 23. Improvement ranging from 16% to 240% can be obtained. Similar results were found when the unloaded edges were restrained and are shown in Fig. 24 for ultimate failure loads and Fig. 25 for the load ratios. The maximum value of ultimate load ðNu Þm and maximum ultimate load ratios, ðRu Þm , are plotted vs. cutout size in Figs. 26 and 27, respectively. There are four points to be made based on these plots: (1) the maximum values correspond to the most severe tailoring in all cases; (2) only slight differences exist between the two cases of free and restrained unload edges; (3) the restrained cases achieve slightly higher ultimate loads for all except the smallest cutout considered; (4) the ultimate loads and load ratios decrease as the cutout becomes larger. All four of these observations are contrary to the trends shown for buckling load as a function of cutout size and degree of tailoring. Therefore, the best design might often be a compromise between the design yielding the highest ultimate failure load and design yielding the highest critical buckling load. In all cases significantly improved structural performance results
Fig. 20. Maximum buckling loads, ðNcr Þm .
Fig. 22. Ultimate loads, Nu .
Fig. 21. Maximum buckling load ratios, ðRcr Þm .
Fig. 23. Ultimate load ratios, Ru .
inplane restraint is added. For example, with a cutout size of 2r=b ¼ 0:4, the optimum b is 0.30 for the free edge and 0.46 for the restrained edge. Finally, a negative effect of tailoring is observed when the plate with smallest cutout is most highly tailored; see the point below the dashed line representing Rcr ¼ 1:0 in Fig. 19. This is the first negative impact discovered for the tailoring concept being investigated and it can be avoided in design by simply using a larger b. To help summarize the above discussion for both side boundary conditions, the maximum values of critical buckling loads, denoted ðNcr Þm , found for any values of the tailoring width ratio are plotted vs. cutout size in Fig. 20. A nonlinear relationship between ðNcr Þm and the cutout size exists for all cases with ðNcr Þm increasing as cutout size increases. If the ðNcr Þm values are normalized by the uniform plate critical buckling loads for each cutout size, the maximum critical load ratios ðRcr Þm are obtained as shown in Fig. 21. Here it can be seen that, relative to buckling, the advantage of tailoring actually increases as the cutout size increases. In addition, the relative benefit of tailoring is much higher for plates with free unloaded edges than with restrained unloaded edges. 4.3. Ultimate loads
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from tailoring, even if a tailored design that is optimal for a certain condition is not the final design selected.
5. Results for cylindrical shells
Fig. 24. Ultimate loads, Nu .
Fig. 25. Ultimate load ratios, Ru .
Tailored cylindrical panels in which the radial deformation was restrained were discussed in Ref. [6]. The panels had a central cutout formed as the intersection of a cylindrical shape and the curved panel. These panels were shown to respond to compressive loads and to develop damage in almost the same way as flat panels with similar restraints. Here the radial deformation restraint is removed except along the unloaded edges. As expected, the response and damage development is quite different for the curved and flat panels in compression. The same material and laminate structure used previously are also used here. Fig. 28 shows plots of the average running load vs. applied end displacement for the uniform shell (b ¼ 1:00), the gently tailored shell (b ¼ 0:64), and the most highly tailored shell (b ¼ 0:10). The cutout size is 2r=b ¼ 0:3. Results for panels obtained without considering damage are shown as dashed lines. The open and filled circles are simply specific points on the curves for which out-of-plane deformations and damage development will be shown in later figures. Some important differences between the curved and flat plates, as well as between uniform and tailored designs, are now discussed. All of the flat plates, uniform or tailored and damaged or undamaged, exhibited a distinct buckling point at a very low load level followed by an extensive postbuckling regime. In addition, no significant damage developed prior to ultimate load and damage initiated well away from the cutout and grew toward the
Fig. 26. Maximum ultimate loads, ðNu Þm .
Fig. 27. Maximum ultimate load ratios, ðRu Þm .
Fig. 28. Average running load Nx vs. end displacement U of cylindrical panels: uniform and two extreme tailoring cases.
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unloaded edges. The uniform and tailored flat plates performed qualitatively in a very similar way but with large quantitative differences. The curved panels performed very differently from the flat panels in almost all of these areas. 5.1. Three specific cases First, consider the response of the undamaged, uniform cylindrical shell shown in Fig. 28. There is substantial nonlinearity in the load–displacement plot almost from the beginning of loading and far prior to damage initiation. This is due to small but increasing out-of-plane deformations and the associated load redistribution around the cutout as the load is increased. However, there is no distinct buckling until the ultimate load is reached at U ¼ 0:0134 in. This ultimate load can be interpreted as a case of limit-point buckling. Load reversal typical of shell buckling is observed at this point and tracing the subsequent ‘‘snap-back’’ in the curve required the use of the Riks solution procedure in ABAQUS. After buckling, a sizable postbuckling load is retained, but this is associated with a very low tangent postbuckling stiffness. The limit point buckling load remains the ultimate load in the undamaged uniform panel over the range of end displacements considered here. The out-of-plane deformations are plotted in Fig. 29. It should be noted that the deformation value associated
with a contour fringe is nearly 20 times smaller than for the contours shown previously for the flat plates. The small initial deformations are outward and at their maximum value at the side of the cutout. These deflections diminish along a circumferential line from the sides of the cutout. The deflection is inward along a longitudinal centerline reaching a maximum at the centerline edge of the cutout. However, at buckling (U ¼ 0:0134 in.), the deflected shape changes dramatically with the largest deformation being inward at a point away from the cutout and both longitudinal and circumferential centerlines. Accounting for damage initiation and progression changes the uniform panel response in many ways. Damage initiates at the side of the cutout in all plies at U ¼ 0:0076 in. as shown in Fig. 30. Fiber damage is present in the 0° and þ45° plies and matrix damage is present in the 90° and 45° plies with essentially the same damage pattern in all cases. Damage progresses in a very narrow circumferential band away from the cutout but never approaches the unloaded edge prior to ultimate failure. The growth of damage is gradual and is associated by a smooth drop in tangent stiffness from load initiation to final failure. The ultimate load is considerably lower than the undamaged panel buckling load. If the growing damage is viewed as a kind of imperfection, the large difference between ultimate load in the damaged and undamaged cases simply demonstrates the imperfection sensitivity typical of curved panels. The
Fig. 29. Radial deformation, uniform shell (b ¼ 1:00, 2r=b ¼ 0:3).
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Fig. 30. Damage progression, uniform shell (b ¼ 1:00, 2r=b ¼ 0:3).
out-of-plane deformations in the damaged panel are also shown in Fig. 29. In this case, the initial shape of the deformations is retained throughout most of the loading history, only increasing in magnitude as the load is increased. Near ultimate load, the narrow circumferential band of failed fibers creates a kind of ‘‘hinge’’ close to the cutout and along this line. This can be seen in the sharp outward peak in the deformed shape at this location and at ultimate load. The maximum deformations are roughly an order of magnitude smaller than in the flat plates. Now examine the response of the undamaged gently tailored shell ( b ¼ 0:64) shown in Fig. 28. Compared to the uniform plate, the buckling load is slightly higher but occurs at a slightly lower critical end displacement. The nonlinearity in the load–displacement curve is still present although less prominent than in the uniform
case. The deformed shapes shown in Fig. 31 show great similarity to those of the uniform panel from the initiation of loading through damage initiation to ultimate load. When damage is accounted for, the ultimate load is only slightly lower than the undamaged panel buckling load. As shown in Fig. 32, damage initiation occurs at the side of the cutout but is delayed substantially compared to the uniform case. This is particularly true for the 0° fiber failure. As in the uniform case, damage growth again occurs in a narrow band away from the side of the cutout and ultimate failure occurs with damage having progressed only a small way across the panel width. This delay in damage initiation and limited growth account for the large increase in ultimate load compared to the uniform panel. The out-of-plane deformation in the damaged case at ultimate load shows a
Fig. 31. Radial deformation, most gently tailored shell (b ¼ 0:64, 2r=b ¼ 0:3).
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Fig. 32. Damage progression, most gently tailored shell (b ¼ 0:64, 2r=b ¼ 0:3).
slight deviation in shape from the earlier deformation shapes indicating that the failure could be a combination of local failures and panel buckling. Finally the tailored panel exhibits less imperfection sensitivity than the uniform panel in that the presence of damage reduces the ultimate load only slightly below the undamaged limit point buckling load. The load–deflection plots for the most highly tailored shell ( b ¼ 0:10) are also shown in Fig. 28. Tailoring has an obvious effect on the undamaged panel buckling behavior in that the snap-back region in the load– deflection curve after buckling is quite sharp and deep. The postbuckling stiffness of the highly tailored panel is much higher than the other two panels shown in the figure and only slightly less than the pre-buckling stiffness. In the absence of damage, the panel could sustain very large postbuckling loads. In the region of the snap-back, the out-of-plane deformation shape changes drastically. Fig. 33 shows deformation plots at five load levels, corresponding to U ¼ 0:0163, 0.0178, 0.0183, 0.0168, and 0.0200 in., that define this snap-back region. Initially the deformations grow in a manner very similar to the uniform and gently tailored panels. The inward deformations away from the cutout and both centerlines are again observed at buckling, however, here they are more sharply concentrated. After buckling and on reloading in the postbuckling range, these inward deflections grow in both magnitude and expanse. When damage is accounted for in the highly tailored case, the defects cause the snap-back region to be eliminated from the load–deflection curve. The response resembles plate buckling with a short postbuckling regime, although the deformed surface plots in Fig. 33 do not show a drastic change in deformed shape at the load level that could be identified as a ‘‘buckling’’ point. The ultimate load is actually slightly greater than the undamaged buckling load in this case. Therefore this panel can be viewed as being much less imperfection sensitive
than either the uniform or the gently tailored panel. The ultimate load and corresponding end displacement is much higher than for these other two panels. The damage patterns are also quite different in this case as shown in Fig. 34. Damage initiates at the side of the cutout and at about the same end displacement as was the case for the gently tailored plate. However, in the current case, damage grows over a much larger portion of the panel and not along the circumferential centerline. The damage growth also takes place over a much larger range of applied end displacement and load magnitude prior to ultimate failure. This indicates that the highly tailored design can produce not only a higher ultimate load but also can give more of a warning prior to ultimate failure when compared to the uniform or gently tailored panels. The analysis procedure was also applied to the same cylindrical panel with four different cutout sizes, with additional values of the tailoring width parameter, and with the circumferential displacements free or restrained on the unloaded edges. Fig. 35 shows the load–deflection plots for all of the free edge cases. A very similar set of curves was obtained for restrained edge cases. Summary results, similar to Figs. 22–27, were distilled from these curves and plotted in Figs. 36–41. No summary plots are shown here for buckling since, when damage was considered, distinct buckling points were not evident. Figs. 36 and 37 summarize the ultimate loads (Nu ) and the corresponding ultimate load ratios (Ru ) for the free unloaded edge condition. Similar results are presented for the restrained unloaded edge condition in Figs. 38 and 39. Relative improvements in ultimate loads range from a low of about 40% to a high of about 155%. The low end of this range is somewhat higher than for the flat panels previously discussed, but the high end of this range is considerably lower than for the flat panels. For all cases, the maximum ultimate loads ðNu Þm and ratios ðRu Þm correspond to the most highly tailored design with b ¼ 0:10. The plot of ðNu Þm as a
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Fig. 33. Radial deformation, most highly tailored shell (b ¼ 0:10, 2r=b ¼ 0:3).
Fig. 34. Damage progression, most highly tailored shell (b ¼ 0:10, 2r=b ¼ 0:3).
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Fig. 35. Average running load Nx vs. end displacement U for cylindrical shells.
Fig. 36. Ultimate loads, Nu .
Fig. 38. Ultimate loads, Nu .
Fig. 37. Ultimate load ratios, Ru .
Fig. 39. Ultimate load ratios, Ru .
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Fig. 40. Maximum ultimate loads, ðNu Þm .
Fig. 41. Maximum ultimate load ratios, ðRu Þm .
function of cutout size presented in Fig. 40 shows that the maximum ultimate loads decrease as the cutout size increases but the maximum ultimate load ratios generally increase as the cutout becomes larger (Fig. 41). The restrained unloaded edge cases are slightly superior to the free edge cases.
6. Conclusions This study has shown that the relatively simple inplane damage analysis described here can do a good job of predicting ultimate failure in flat and curved composite panels with cutouts. Test results were closely matched using material properties and experimental data from outside sources [9,10]. These tests included a flat plate having a buckling load much lower than the ultimate load and a curved panel having an ultimate load very close to the theoretical buckling load. A simple tailoring concept was shown to be very effective in increasing compressive buckling loads and ultimate loads for flat plates with a central cutout. Effects of cutout size, value of the tailoring width parameter, and inplane restraint on the unloaded edges were investigated. Optimal tailoring produced relative im-
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provements ranging from 40% to 175% in buckling load and 190–240% in ultimate load capacity when compared to uniform plates with the same cutout sizes. Results presented previously [6] for the same flat plates, but with all out-of-plane deformations being artificially suppressed, showed optimal improvements in ultimate load ranging from 100% to 165%. Comparing these two sets of results shows that the value of tailoring is much higher in thin flat plates where the failure mechanism is complex due to its occurrence within the postbuckling regime. In addition to these large increases in the buckling and ultimate loads, the same tailoring approach provides similar improvements in the postbuckling stiffness of the panel. This can be of special importance when the postbuckled plate is used as a component of a stiffened panel. Damage initiation was observed to occur away from the cutout and along a circumferential centerline. Damage did not progress over the entire length of this centerline prior to ultimate failure, which was rather abrupt in all cases whether uniform or tailored. The one disappointing result is that the tailoring concept did not prove to act as a damage arrest mechanism in the plates in compression, regardless of whether they are allowed to buckle or whether out-of-plane deformations are restrained. This arrest feature was found to be present when the loads are tensile [6]. The same damage analyses were conducted for panels identical to the flat plates but having a cylindrical shape representing 1/4 of a complete cylinder. No distinct buckling prior to ultimate failure was observed. Tailoring proved to lower the imperfection (local damage) sensitivity of the curved panels when compared to the load–deflection plots of undamaged panels. In fact, in some cases the tailored panel ultimate loads slightly exceeded the theoretical undamaged buckling loads. To the contrary, the ultimate load for the uniform curved panel was found to be much lower than the undamaged buckling load. Relative improvements in ultimate loads ranged from a low of about 40% to a high of about 155%. In the curved panels, damage initiated at the side of the cutout and progressed in a narrow circumferential band (uniform and gently tailored cases) or along an inclined band in the highly tailored cases. Failure occurred after the damage had progressed only part way across the panel, with the extent of the damaged area being larger for the more highly tailored panels. The large difference in the damage initiation and progression between the flat and the curve panels is of some interest. The curved panel damage strongly resembled the previously investigated flat panel damage where outof-plane deformations were prevented [6]. It is believed that the panel curvature provides so much stiffness against these out-of-plane deformations as to make them respond in a way similar to the artificially restrained flat panels.
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The final conclusion drawn from this work, in conjunction with the previous study [6], is that the tailoring concept investigated here can provide excellent improvements in ultimate load capacity in flat and curved with the largest benefits occurring in thin flat panels that are loaded far into the compressive postbuckling regime. Acknowledgements Financial support for this work was provided by NASA, the State of South Carolina, and Clemson University through the EPSCoR grant ‘‘Development and Enhancement of Research Capability for Aircraft Structures and Materials’’. References [1] Biggers SB, Srinivasan S. Compression buckling response of tailored composite plates. AIAA J 1993;31:590–6. [2] Biggers SB, Pageau S. Shear buckling response of tailored composite plates. AIAA J 1994;32:1100–3.
[3] Biggers Jr SB, Browder Jr TM. Buckling-load interaction in tailored composite plates. Compos Eng 1994;4:745–61. [4] Biggers SB, Srinivasan S. Postbuckling response of piece-wise uniform tailored composite plates in compression. J Reinf Plast Compos 1994;13(9):803–21. [5] Baranski AT, Biggers Jr SB. Post-buckling analysis of tailored composite plates with progressive damage. Compos Struct 1999; 46:245–55. [6] Biggers Jr SB, Xie D. Damage progression analysis in tailored laminated plates and shells with a cutout. In: Proceedings of the American Society for Composites, 16th Annual Technical Conference. ASC Paper 136, Blacksburg, VA, September 2001. [7] HKS, Inc. ABAQUS Theory and UserÕs Manual, version 5.8. HKS, Pawtucket, RI. [8] Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I––Analysis. J Compos Mater 1991;25:2–43. [9] Starnes Jr JH, Rouse M. Postbuckling and failure characteristics of selected flat rectangular graphite-epoxy plates loaded in compression. AIAA Paper 81-0543, April 1981. [10] Knight Jr NF, Starnes Jr JH. Postbuckling behavior of axially compressed graphite-epoxy cylindrical panels with circular holes. ASME J Press Vessel Technol 1985;107:394–402. [11] Engelstad SP, Reddy JN, Knight Jr NF. Postbuckling response and failure prediction of graphite-epoxy plates loaded in compression. AIAA J 1992;38:2106–13.
Postbuckling analysis with progressive damage modeling in tailored laminated plates and shells with a cutout De Xie, Sherrill B. Biggers Jr.
*
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA
Abstract An approach to modeling inplane damage progression in postbuckled laminated composite panels is shown to be accurate by comparison to experimental test data from other sources. A simple tailoring concept is shown to be very effective in increasing compressive buckling loads and ultimate loads for flat plates and curved panels with a central cutout. Effects of cutout size, the degree of tailoring, and inplane restraint on the unloaded edges are investigated. Optimal tailoring produces relative improvements in the flat plates ranging from 40% to 175% in buckling load and 190–240% in ultimate load capacity when compared to uniform plates with the same cutout sizes. In the curved panels, tailoring lowers the imperfection sensitivity and in some cases produces ultimate loads greater than the theoretical undamaged buckling loads. To the contrary, the ultimate load for the uniform curved panel is much lower than the undamaged buckling load. Relative improvements in ultimate loads range from a low of about 40% to a high of about 155% compared to uniform curved panels. Large differences in the damage initiation locations and damage progression patterns are shown between the flat and the curved panels. In summary, the tailoring concept investigated here can provide excellent improvements in ultimate load capacity in flat and curved panels with the largest benefits occurring in thin flat panels that are loaded far into the compressive postbuckling regime. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Damage progression; Postbuckling; Laminated plates and shells; Tailored design; Finite element analysis
1. Introduction Composite materials offer a natural opportunity to enhance structural integrity through the use of various laminate tailoring concepts. Among these is a simple stiffness tailoring concept which simply removes all axially oriented 0° materials from the central region of a rectangular panel and relocates the material in the edge regions of the panel. This approach has been shown to improve buckling loads, initial failure loads, and residual strength in laminated panels without affecting their weight in any way. The uniaxial stiffness tailoring concept described above was studied as a way to increase the buckling resistance of rectangular composite plates under uniaxial compression and increases of 200% or more were shown to be possible when compared to uniform plates [1]. A similar tailoring concept was investigated for shearing loads and buckling load improvements of up to
*
Corresponding author. Fax: +1-864-656-4435. E-mail address: [email protected] (S.B. Biggers Jr.).
86% were achieved [2]. Furthermore, this approach was investigated for improved buckling resistance of rectangular composite plates subject to general in-plane combined axial, transverse and shear loadings [3]. Very large increases in buckling loads were found due to tailoring when transverse tensile loads are present along with axial compression. In many practical cases, panels are allowed to enter the postbuckling regime and operate at load and strain levels exceeding critical buckling values. Due to the higher load levels and the more complex structural response in the postbuckling regime, damage may initiate and propagate to some extent in panels and finally make them fail due to a combination of effects. Therefore a damage modeling scheme is a necessary component in the postbuckling analysis procedure. The previously mentioned research on initial buckling was extended to postbuckling with first-ply failure identified by the Tsai– Hill criterion [4]. This study showed that the compressive postbuckling stiffness and first-ply failure loads could also be greatly improved by tailoring. This work was further extended to consider the response after firstply failure in which buckling initiation, postbuckling
0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 2 3 3 - 7
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damage growth and ultimate failure were considered for rectangular plates in compression [5]. Damage growth in tailored flat plates and cylindrical shells containing a central cutout of varying size has also been investigated [6]. To simplify the structural response, this last work restricted the deformation to occur within the plane of the panels. In the case of tensile loading, the tailored region was found to arrest a temporarily unstable progression of damage. The current study releases the restraints on the outof-plane deformation in Ref. [6] and is a natural extension of the work in Ref. [5]. The damage is still assumed to occur within the plane of panels but now bending and twisting associated with the out-of-plane response also contribute to the damage development. No delamination or transverse damage mode is included. The analysis method was validated by comparison to experimental data from tests conducted by others. Results are presented for a set of flat and curved plates having a range of cutout sizes. In each case, different degrees of tailoring were also investigated so that the interaction with panel curvature, cutout size, and extent of tailoring could be evaluated. As expected, the flat plates were found to behave quite differently from the curved panels. This is contrary to the results presented in Ref. [6] where out-of-plane deformations were artificially restrained.
2. Tailoring approach A typical panel evaluated in this study is shown in Fig. 1. In the limit as the included angle approaches zero, the panel becomes a flat plate. The curved panel geometry considered in this study is a cylindrical shell with an included angle h ¼ 90°. Compressive loading was taken to be uniaxial in the x-direction and applied as a uniform end displacement. The panel was clamped on the loaded end and was simply supported on the
Fig. 1. Geometry of uniform and tailored panels.
Table 1 Mechanical properties of IM7/8551-7 and T300/5208 E11 (Msi) E22 (Msi) G12 (Msi) m12 XT (Ksi) XC (Ksi) YT (Ksi) YC (Ksi) S (Ksi)
IM7/8551-7
T300/5208
18.9 1.36 0.653 0.33 366.4 132.9 10.7 10.7 17.4
19.0 1.89 0.93 0.38 200.0 165.0 11.74 27.41 10.0
unloaded edges to prevent the panel from buckling as a wide column. Circumferential displacements on the unloaded edges were assumed to be either free or completely restrained, corresponding to upper and lower bounds that might be encountered in practical applications. The tailoring concept investigated here is a simple repositioning of all axially oriented 0° material into regions of certain widths near the edges of the panel. The baseline panel is square and has a uniform quasiisotropic layup [45/0/90]S with mechanical properties typical of IM7/8551-7 prepreg material (Table 1). Properties are also given for T300/5208 prepreg used for comparison to existing experimental data. The edge regions in the tailored panels are of total width b1 while the overall panel width (arc length) is b. The 0° material is uniformly distributed in the edge regions. The width ratio, b ¼ b1 =b, defines the extent of the tailoring and when b ¼ 1:0, the panel is uniform. When b < 1:0, the total thickness of the 0° material in the edge regions is increased such that the total weight of material in the panel remains unchanged.
3. Analysis method The analysis was conducted using the ABAQUS finite element code [7]. Only in-plane damage was taken into account, i.e. no interlaminar damage such as delamination was considered. The damage evolution model used is a simple ply-discount method in which stress and stiffness are reduced locally and instantaneously to zero once failure occurs at a point in a ply. The nonlinear behavior of the inplane shear modulus suggested by Chang [8] was also incorporated. The effect of this was to account for the slight softening of the shear stiffness as the stresses increased. The UMAT routine within ABAQUS was used to account for nonlinearities in the material due to local failures and softening of the shear modulus. These changes due to damage were introduced at individual integrations points as they occurred during loading. The Hashin failure criteria, as modified by Chang [8], were used to indicate damage initiation. To
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initiate out-of-plane deformation, an out-of-plane imperfection equal 1% of the panel thickness was introduced uniformly along the edge of the cutout. To demonstrate the accuracy of the method, the results predicted by the present analysis method were compared with the experimental results provided by Starnes and Rouse [9] for the flat plate (H4) and by Knight and Starnes [10] for the cylindrical shell (CP8). The geometry and finite element meshes of the plate and the shell are shown in Fig. 2. Both the flat plate (H4) and the cylindrical panel (CP8) were fabricated from unidirectional T300/5208 prepreg. Table 1 gives the mechanical properties except that E11 ¼ 19:6 Msi was used for panel CP8 as given in Ref. [10]. The loading and boundary conditions used for the analysis were selected to be consistent with the experimental tests. For the flat plate H4, the ply thickness is 0.00574 in. and the laminate stacking sequence is [45/0/90]3S . Engelstad et al. [11] also performed a similar progressive damage analysis on the plate. Fig. 3 compares the end reaction vs. end shortening curves obtained from present analysis with that from the experiment, and shows good correlation between the two. The plate supports increasing load after buckling until sufficient damage occurs to make it fail completely. The ultimate load predicted is within 4% of the ultimate test load and the corresponding strain is within about 7% of the test strain. The mode shapes at initial buckling (A) and ultimate load (B) shown in Fig. 4 agree with those given by Engelstad et al. [11] The same analysis procedure was applied to the CP8 cylindrical shell that has a ply thickness of 0.0056 in. and a laminate stacking sequence of [45/90/02 /90/45]S .
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Fig. 3. Comparison of end reaction vs. end shortening for H4, experimental [9] and analytical.
Fig. 4. Mode shapes for H4.
Fig. 2. Geometry and finite element meshes of plate H4 and shell CP8.
The predicted end reaction vs. end shortening curve is compared with the experimental data in Fig. 5. The analysis without considering damage shows a sharp drop in load after buckling, consistent with shell buckling. The damage analysis predicts damage development prior to, and an ultimate load slightly lower than, the theoretical buckling point. The damage analysis and experimental test results are remarkably similar. This is despite the fact that some delamination was observed at the free edge of the cutout in the physical test but was not taken into account in present analysis. The only major difference between the damage analysis and the test data is the much lower postfailure residual strength in the test, probably due to significant edge delamination after ultimate load. Fig. 6 shows the mode shapes at ultimate load (A) and a postfailure load (B). These deformation patterns agree well with those obtained by Knight and Starnes [10].
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Fig. 5. Comparison of end reaction vs. end shortening for CP8, experimental [10] and analytical.
Fig. 7. Finite element mesh for a typical flat plate.
Fig. 6. Mode shapes for CP8.
These comparisons of the analytical results to two rather different experimental test cases give sufficient confidence in the damage analysis to apply it to other cases in determining buckling and ultimate loads in the presence of inplane damage. This is done below, first for tailored flat plates with a central cutout and then for cylindrical shells.
4. Results for flat plates Now the simple tailoring concept described previously will be investigated for its effectiveness in increasing buckling and ultimate loads and also in controlling damage growth in flat plates with a central cutout. A typical finite element model for a uniform plate is shown in Fig. 7. Models for the tailored plates were similar. All plates are 6 in. by 6 in. and the uniform end displacement U is applied equally at each end of the
Fig. 8. Average running load Nx vs. end displacement U of flat plates: uniform and two extreme tailoring cases.
plate. Fig. 8 shows the average running load vs. applied end displacement curves for the uniform plate and two tailoring cases: the most highly tailored (b ¼ 0:10) plate and the most gently tailored (b ¼ 0:64) plate. The cutout size is 2r=b ¼ 0:3. Similar results obtained for other cutout sizes and other tailoring parameters will be shown later. The panel responses neglecting damage are also shown in Fig. 8 with dashed lines. The filled circles on the curves indicate load points for which deformed shapes will be shown in later figures. The Newton– Raphson method was used to perform this nonlinear analysis without any difficulties in convergence. Some important features can be observed in Fig. 8. Regardless of the specific tailoring, a well-defined buckling point is observed followed by a postbuckling
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response with reduced stiffness. The much higher postbuckling stiffness in the highly tailored plate follows the same trend found in earlier studies on plates without the central cutout [5]. The same can be said for the much higher ultimate load and corresponding overall panel strain. However for all three cases, the nearly complete coincidence of the curves with and without damage up until ultimate load suggests a sudden failure with no significant gradual damage progression prior to final failure. This is quite different from the tension load case considered in Ref. [6] where it was found that tailoring could arrest damage growth to a certain extent depending on the tailoring parameter. These observations can be better understood by examining the pictures of the deformed plates and the damage patterns presented below. 4.1. Three specific cases First, consider the uniform plate. Fig. 9 shows a sequence of deformed 1/4-plate models at load levels corresponding to the filled circles on the curve shown in Fig. 8. Deformed configurations are shown in an oblique view and scaled by a factor of 5 to exaggerate the deformation. Contours of out-of-plane displacement are cast on the deformed configurations using a fixed contour level with the same maximum and minimum value
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for all plots. The uniform flat plate was found (Fig. 8) to buckle at an end shortening U ¼ 0:0013 in. and this is reflected in the distinct geometry change around the cutout from U ¼ 0:0012 to 0.0013 in. The initial buckling mode shape is a single half wave over the entire plate. Fig. 9 also shows the development of the plate deformation during the postbuckling stage. The upward deformation around the cutout spreads mainly in the direction transverse to the load direction. The deflected shape becomes flatter near the center as the load and magnitude of the deformation increase, just as is the case in plates without a cutout. At an end shortening U approaching 0.0075 in., a downward deformation initiates away from the cutout, indicating a mode shift from one to three half waves along the load direction. The combination of these effects creates high bending and twisting gradients near the center of the unloaded edge. At the same time the large out-of-plane bending deformation around the cutout reduces the axial stiffness of the central region of the plate and makes this region shed compressive load toward the side regions. Therefore, the region near the center of the unloaded edges has the highest combination of inplane, bending, and twisting loads and damage initiates in this area. This combination of effects is in fact one of the reasons that the tailoring concept investigated here should prove to be effective.
Fig. 9. Out-of-plane displacements, uniform plate (b ¼ 1:00, 2r=b ¼ 0:3).
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Fig. 10. Damage, uniform plate just after ultimate load (b ¼ 1:00, 2r=b ¼ 0:3).
Fig. 10 presents plots showing both the matrix failure and the fiber failure in top four plies just after the ultimate load condition. The dark areas are the damaged areas. Based on the observations of the out-of-plane deformations discussed above, it is easy to see why the damage exists in the regions shown rather than at the sides of the cutouts as one might expect for purely inplane response. In general, the matrix failure of the þ45° ply is similar to fiber failure of the 45° ply and vice versa. (Note: the þ45° fibers run tangent to the side of the cutout in the quadrant of the plate shown here.) The same can be said for the 0° ply and the 90° ply. Although nonlinearity in the matrix shear modulus started devel-
oping at loads much less than ultimate, this did not have a significant effect on the overall response. It was observed that failures in the matrix and fiber started developing from a load very close to the ultimate and quickly propagated to the distribution shown in Fig. 10. This reinforces the idea of sudden failure mentioned previously. Similar deformation and damage developments were observed for the two tailored plates. Figs. 11 and 12 show the out-of-plane deformation configurations and the damage patterns, respectively, for the most gently tailored plate. In this case (b ¼ 0:64) the 0° material is broadly distributed from near the edge of the cutout to
Fig. 11. Out-of-plane deformation, most gently tailored plate (b ¼ 0:64, 2r=b ¼ 0:3).
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Fig. 12. Damage, most gently tailored plate just after ultimate load (b ¼ 0:64, 2r=b ¼ 0:3).
the unloaded edge. The dashed line marks the inner edge of the 0° material. This gentle tailoring greatly reduces the inplane stress concentration at the cutout and slightly increases the stiffnesses near the unloaded edge. However, since the cutout stress concentration did not prove to be crucial in the uniform plate, it is understandable that this gently tailored plate performs very much in the same way as the uniform plate. The buckling load is slightly increased as are the ultimate load and ultimate end displacement. But the out-of-plane displacements and the damage development patterns are nearly identical to those of the uniform plate.
Figs. 13 and 14 show deformations and damage patterns for the most highly tailored plate (b ¼ 0:10). Here the 0° material is concentrated in a narrow band near the unloaded edge. This localization of material may or may not be practical, depending on the application. The deformations appear quite similar to the other two cases except for the increased resistance to rotation near the unloaded edge. This tends to force the critical location and damage locations away from the unloaded edge to some degree. The main effect is to delay the initiation of fiber damage until a much higher applied end displacement and to retain a much higher
Fig. 13. Out-of-plane deformation, most highly tailored plate (b ¼ 0:10, 2r=b ¼ 0:3).
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Fig. 14. Damage, most highly tailored plate just after ultimate load (b ¼ 0:10, 2r=b ¼ 0:3).
postbuckling stiffness in the panel. These effects combine to give a greatly increased ultimate load. However, even this highly tailored design does not act to arrest damage growth and make the failure less sudden. Compared to the uniform plate, the highly tailored design is found to increase the postbuckling stiffness by 179%, the ultimate strength by 229%, and the ultimate axial strain by 39%. The corresponding improvements for the gently tailored design are 21%, 29%, and 5%. Furthermore, despite making the plate thinner in the center region, neither tailored design has higher outof-plane deformations than the uniform plate at a given end displacement. For the uniform plate at U ¼ 0:0125 in., the peak value for the upward and downward deformations are 0.137 and 0.0510 in., respectively. For the highly tailored plate, the corresponding values are 0.120 and 0.0547 in., while for the gently tailored plate the corresponding values are 0.124 and 0.0550 in. Therefore, at least for the cutout size considered above, both the most highly tailored and the most gently tailored designs offer improvement in every aspect of structural performance. Unfortunately, however, the desirable trait of damage arrest that was observed in Ref. [6] for tailored plates having a cutout and tensile loading does not present itself in the case of thin compressively loaded tailored plates. To give a more complete understanding of the effects highlighted above for three specific cases, now consider the response over a range of cutout sizes and a wider range of tailoring parameters. The average running load vs. applied end displacement curves are shown in Fig. 15 for the flat plates with four different cutout sizes. Results are shown for uniform ( b ¼ 1:0) plates and for tailored plates with b ranging from 0.10 to as high as 0.76 depending on the cutout size. Here it can be seen that the same trends discussed previously hold for the full range of parameters. Now the buckling loads and the ultimate loads obtained from this figure will be plotted vs. the tailoring width parameter and cutout size to further aid
our understanding of the effectiveness of tailoring on structural performance. 4.2. Initial buckling loads Fig. 16 shows the critical buckling loads (Ncr ) for each cutout size as a function of the tailoring width ratio ( b). Values for the uniform plates are also shown as separate points at b ¼ 1:0 for reference. These data are also shown as buckling load ratios (Rcr ) in Fig. 17 where the uniform plate critical buckling loads for each cutout size were used in the normalization. These plots show that tailoring can increase the critical buckling loads in a range from 24% to 175% with larger percentage improvements occurring with the larger cutouts and for intermediate values of the tailoring width ratio. As an example, with 2r=b ¼ 0:3, the optimum tailoring is with b ¼ 0:4 producing a 120% increase in buckling load (Rcr ¼ 2:20). Simulations were also performed for flat plates for which circumferential displacements on the unloaded edges were restrained. The effect of this restraint is to introduce a nonuniform transverse compressive load along the previously unloaded edges. Boundary conditions in most practical cases would be between the free and totally restrained cases considered here. Load vs. displacement plots very similar in character to those shown in Fig. 15 were obtained and they are not repeated here. However, some results are presented in Figs. 18 and 19, which summarize the buckling loads and corresponding Rcr values for the restrained edge cases. Compared to the free edge case, the following differences were found. First, the maximum value of critical buckling loads and Rcr values are lower for all cutout sizes. This is reasonable since the panel is effectively now loaded in biaxial compression. Second, the optimum design for buckling tends to shift from an intermediate degree of tailoring towards slightly gentler tailoring, i.e. the optimum b increases when the side
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Fig. 15. Average running load Nx vs. end displacement U of flat plates.
Fig. 18. Critical buckling loads, Ncr . Fig. 16. Critical buckling loads, Ncr .
Fig. 17. Critical buckling load ratios, Rcr .
Fig. 19. Critical buckling load ratios, Rcr .
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Next, consider the effect that tailoring has on the ultimate failure load. The peak values taken from each
load displacement curve in Fig. 15 are plotted against b in Fig. 22. For all cutout sizes, the maximum ultimate load is obtained for b ¼ 0:1. The same observation was reported for postbuckled plates without cutouts in Ref. [5]. The data in Fig. 22 were normalized by the ultimate loads of the uniform plate with the same size cutout and these ultimate load ratios ðRu Þ are plotted in Fig. 23. Improvement ranging from 16% to 240% can be obtained. Similar results were found when the unloaded edges were restrained and are shown in Fig. 24 for ultimate failure loads and Fig. 25 for the load ratios. The maximum value of ultimate load ðNu Þm and maximum ultimate load ratios, ðRu Þm , are plotted vs. cutout size in Figs. 26 and 27, respectively. There are four points to be made based on these plots: (1) the maximum values correspond to the most severe tailoring in all cases; (2) only slight differences exist between the two cases of free and restrained unload edges; (3) the restrained cases achieve slightly higher ultimate loads for all except the smallest cutout considered; (4) the ultimate loads and load ratios decrease as the cutout becomes larger. All four of these observations are contrary to the trends shown for buckling load as a function of cutout size and degree of tailoring. Therefore, the best design might often be a compromise between the design yielding the highest ultimate failure load and design yielding the highest critical buckling load. In all cases significantly improved structural performance results
Fig. 20. Maximum buckling loads, ðNcr Þm .
Fig. 22. Ultimate loads, Nu .
Fig. 21. Maximum buckling load ratios, ðRcr Þm .
Fig. 23. Ultimate load ratios, Ru .
inplane restraint is added. For example, with a cutout size of 2r=b ¼ 0:4, the optimum b is 0.30 for the free edge and 0.46 for the restrained edge. Finally, a negative effect of tailoring is observed when the plate with smallest cutout is most highly tailored; see the point below the dashed line representing Rcr ¼ 1:0 in Fig. 19. This is the first negative impact discovered for the tailoring concept being investigated and it can be avoided in design by simply using a larger b. To help summarize the above discussion for both side boundary conditions, the maximum values of critical buckling loads, denoted ðNcr Þm , found for any values of the tailoring width ratio are plotted vs. cutout size in Fig. 20. A nonlinear relationship between ðNcr Þm and the cutout size exists for all cases with ðNcr Þm increasing as cutout size increases. If the ðNcr Þm values are normalized by the uniform plate critical buckling loads for each cutout size, the maximum critical load ratios ðRcr Þm are obtained as shown in Fig. 21. Here it can be seen that, relative to buckling, the advantage of tailoring actually increases as the cutout size increases. In addition, the relative benefit of tailoring is much higher for plates with free unloaded edges than with restrained unloaded edges. 4.3. Ultimate loads
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from tailoring, even if a tailored design that is optimal for a certain condition is not the final design selected.
5. Results for cylindrical shells
Fig. 24. Ultimate loads, Nu .
Fig. 25. Ultimate load ratios, Ru .
Tailored cylindrical panels in which the radial deformation was restrained were discussed in Ref. [6]. The panels had a central cutout formed as the intersection of a cylindrical shape and the curved panel. These panels were shown to respond to compressive loads and to develop damage in almost the same way as flat panels with similar restraints. Here the radial deformation restraint is removed except along the unloaded edges. As expected, the response and damage development is quite different for the curved and flat panels in compression. The same material and laminate structure used previously are also used here. Fig. 28 shows plots of the average running load vs. applied end displacement for the uniform shell (b ¼ 1:00), the gently tailored shell (b ¼ 0:64), and the most highly tailored shell (b ¼ 0:10). The cutout size is 2r=b ¼ 0:3. Results for panels obtained without considering damage are shown as dashed lines. The open and filled circles are simply specific points on the curves for which out-of-plane deformations and damage development will be shown in later figures. Some important differences between the curved and flat plates, as well as between uniform and tailored designs, are now discussed. All of the flat plates, uniform or tailored and damaged or undamaged, exhibited a distinct buckling point at a very low load level followed by an extensive postbuckling regime. In addition, no significant damage developed prior to ultimate load and damage initiated well away from the cutout and grew toward the
Fig. 26. Maximum ultimate loads, ðNu Þm .
Fig. 27. Maximum ultimate load ratios, ðRu Þm .
Fig. 28. Average running load Nx vs. end displacement U of cylindrical panels: uniform and two extreme tailoring cases.
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unloaded edges. The uniform and tailored flat plates performed qualitatively in a very similar way but with large quantitative differences. The curved panels performed very differently from the flat panels in almost all of these areas. 5.1. Three specific cases First, consider the response of the undamaged, uniform cylindrical shell shown in Fig. 28. There is substantial nonlinearity in the load–displacement plot almost from the beginning of loading and far prior to damage initiation. This is due to small but increasing out-of-plane deformations and the associated load redistribution around the cutout as the load is increased. However, there is no distinct buckling until the ultimate load is reached at U ¼ 0:0134 in. This ultimate load can be interpreted as a case of limit-point buckling. Load reversal typical of shell buckling is observed at this point and tracing the subsequent ‘‘snap-back’’ in the curve required the use of the Riks solution procedure in ABAQUS. After buckling, a sizable postbuckling load is retained, but this is associated with a very low tangent postbuckling stiffness. The limit point buckling load remains the ultimate load in the undamaged uniform panel over the range of end displacements considered here. The out-of-plane deformations are plotted in Fig. 29. It should be noted that the deformation value associated
with a contour fringe is nearly 20 times smaller than for the contours shown previously for the flat plates. The small initial deformations are outward and at their maximum value at the side of the cutout. These deflections diminish along a circumferential line from the sides of the cutout. The deflection is inward along a longitudinal centerline reaching a maximum at the centerline edge of the cutout. However, at buckling (U ¼ 0:0134 in.), the deflected shape changes dramatically with the largest deformation being inward at a point away from the cutout and both longitudinal and circumferential centerlines. Accounting for damage initiation and progression changes the uniform panel response in many ways. Damage initiates at the side of the cutout in all plies at U ¼ 0:0076 in. as shown in Fig. 30. Fiber damage is present in the 0° and þ45° plies and matrix damage is present in the 90° and 45° plies with essentially the same damage pattern in all cases. Damage progresses in a very narrow circumferential band away from the cutout but never approaches the unloaded edge prior to ultimate failure. The growth of damage is gradual and is associated by a smooth drop in tangent stiffness from load initiation to final failure. The ultimate load is considerably lower than the undamaged panel buckling load. If the growing damage is viewed as a kind of imperfection, the large difference between ultimate load in the damaged and undamaged cases simply demonstrates the imperfection sensitivity typical of curved panels. The
Fig. 29. Radial deformation, uniform shell (b ¼ 1:00, 2r=b ¼ 0:3).
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Fig. 30. Damage progression, uniform shell (b ¼ 1:00, 2r=b ¼ 0:3).
out-of-plane deformations in the damaged panel are also shown in Fig. 29. In this case, the initial shape of the deformations is retained throughout most of the loading history, only increasing in magnitude as the load is increased. Near ultimate load, the narrow circumferential band of failed fibers creates a kind of ‘‘hinge’’ close to the cutout and along this line. This can be seen in the sharp outward peak in the deformed shape at this location and at ultimate load. The maximum deformations are roughly an order of magnitude smaller than in the flat plates. Now examine the response of the undamaged gently tailored shell ( b ¼ 0:64) shown in Fig. 28. Compared to the uniform plate, the buckling load is slightly higher but occurs at a slightly lower critical end displacement. The nonlinearity in the load–displacement curve is still present although less prominent than in the uniform
case. The deformed shapes shown in Fig. 31 show great similarity to those of the uniform panel from the initiation of loading through damage initiation to ultimate load. When damage is accounted for, the ultimate load is only slightly lower than the undamaged panel buckling load. As shown in Fig. 32, damage initiation occurs at the side of the cutout but is delayed substantially compared to the uniform case. This is particularly true for the 0° fiber failure. As in the uniform case, damage growth again occurs in a narrow band away from the side of the cutout and ultimate failure occurs with damage having progressed only a small way across the panel width. This delay in damage initiation and limited growth account for the large increase in ultimate load compared to the uniform panel. The out-of-plane deformation in the damaged case at ultimate load shows a
Fig. 31. Radial deformation, most gently tailored shell (b ¼ 0:64, 2r=b ¼ 0:3).
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Fig. 32. Damage progression, most gently tailored shell (b ¼ 0:64, 2r=b ¼ 0:3).
slight deviation in shape from the earlier deformation shapes indicating that the failure could be a combination of local failures and panel buckling. Finally the tailored panel exhibits less imperfection sensitivity than the uniform panel in that the presence of damage reduces the ultimate load only slightly below the undamaged limit point buckling load. The load–deflection plots for the most highly tailored shell ( b ¼ 0:10) are also shown in Fig. 28. Tailoring has an obvious effect on the undamaged panel buckling behavior in that the snap-back region in the load– deflection curve after buckling is quite sharp and deep. The postbuckling stiffness of the highly tailored panel is much higher than the other two panels shown in the figure and only slightly less than the pre-buckling stiffness. In the absence of damage, the panel could sustain very large postbuckling loads. In the region of the snap-back, the out-of-plane deformation shape changes drastically. Fig. 33 shows deformation plots at five load levels, corresponding to U ¼ 0:0163, 0.0178, 0.0183, 0.0168, and 0.0200 in., that define this snap-back region. Initially the deformations grow in a manner very similar to the uniform and gently tailored panels. The inward deformations away from the cutout and both centerlines are again observed at buckling, however, here they are more sharply concentrated. After buckling and on reloading in the postbuckling range, these inward deflections grow in both magnitude and expanse. When damage is accounted for in the highly tailored case, the defects cause the snap-back region to be eliminated from the load–deflection curve. The response resembles plate buckling with a short postbuckling regime, although the deformed surface plots in Fig. 33 do not show a drastic change in deformed shape at the load level that could be identified as a ‘‘buckling’’ point. The ultimate load is actually slightly greater than the undamaged buckling load in this case. Therefore this panel can be viewed as being much less imperfection sensitive
than either the uniform or the gently tailored panel. The ultimate load and corresponding end displacement is much higher than for these other two panels. The damage patterns are also quite different in this case as shown in Fig. 34. Damage initiates at the side of the cutout and at about the same end displacement as was the case for the gently tailored plate. However, in the current case, damage grows over a much larger portion of the panel and not along the circumferential centerline. The damage growth also takes place over a much larger range of applied end displacement and load magnitude prior to ultimate failure. This indicates that the highly tailored design can produce not only a higher ultimate load but also can give more of a warning prior to ultimate failure when compared to the uniform or gently tailored panels. The analysis procedure was also applied to the same cylindrical panel with four different cutout sizes, with additional values of the tailoring width parameter, and with the circumferential displacements free or restrained on the unloaded edges. Fig. 35 shows the load–deflection plots for all of the free edge cases. A very similar set of curves was obtained for restrained edge cases. Summary results, similar to Figs. 22–27, were distilled from these curves and plotted in Figs. 36–41. No summary plots are shown here for buckling since, when damage was considered, distinct buckling points were not evident. Figs. 36 and 37 summarize the ultimate loads (Nu ) and the corresponding ultimate load ratios (Ru ) for the free unloaded edge condition. Similar results are presented for the restrained unloaded edge condition in Figs. 38 and 39. Relative improvements in ultimate loads range from a low of about 40% to a high of about 155%. The low end of this range is somewhat higher than for the flat panels previously discussed, but the high end of this range is considerably lower than for the flat panels. For all cases, the maximum ultimate loads ðNu Þm and ratios ðRu Þm correspond to the most highly tailored design with b ¼ 0:10. The plot of ðNu Þm as a
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Fig. 33. Radial deformation, most highly tailored shell (b ¼ 0:10, 2r=b ¼ 0:3).
Fig. 34. Damage progression, most highly tailored shell (b ¼ 0:10, 2r=b ¼ 0:3).
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Fig. 35. Average running load Nx vs. end displacement U for cylindrical shells.
Fig. 36. Ultimate loads, Nu .
Fig. 38. Ultimate loads, Nu .
Fig. 37. Ultimate load ratios, Ru .
Fig. 39. Ultimate load ratios, Ru .
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Fig. 40. Maximum ultimate loads, ðNu Þm .
Fig. 41. Maximum ultimate load ratios, ðRu Þm .
function of cutout size presented in Fig. 40 shows that the maximum ultimate loads decrease as the cutout size increases but the maximum ultimate load ratios generally increase as the cutout becomes larger (Fig. 41). The restrained unloaded edge cases are slightly superior to the free edge cases.
6. Conclusions This study has shown that the relatively simple inplane damage analysis described here can do a good job of predicting ultimate failure in flat and curved composite panels with cutouts. Test results were closely matched using material properties and experimental data from outside sources [9,10]. These tests included a flat plate having a buckling load much lower than the ultimate load and a curved panel having an ultimate load very close to the theoretical buckling load. A simple tailoring concept was shown to be very effective in increasing compressive buckling loads and ultimate loads for flat plates with a central cutout. Effects of cutout size, value of the tailoring width parameter, and inplane restraint on the unloaded edges were investigated. Optimal tailoring produced relative im-
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provements ranging from 40% to 175% in buckling load and 190–240% in ultimate load capacity when compared to uniform plates with the same cutout sizes. Results presented previously [6] for the same flat plates, but with all out-of-plane deformations being artificially suppressed, showed optimal improvements in ultimate load ranging from 100% to 165%. Comparing these two sets of results shows that the value of tailoring is much higher in thin flat plates where the failure mechanism is complex due to its occurrence within the postbuckling regime. In addition to these large increases in the buckling and ultimate loads, the same tailoring approach provides similar improvements in the postbuckling stiffness of the panel. This can be of special importance when the postbuckled plate is used as a component of a stiffened panel. Damage initiation was observed to occur away from the cutout and along a circumferential centerline. Damage did not progress over the entire length of this centerline prior to ultimate failure, which was rather abrupt in all cases whether uniform or tailored. The one disappointing result is that the tailoring concept did not prove to act as a damage arrest mechanism in the plates in compression, regardless of whether they are allowed to buckle or whether out-of-plane deformations are restrained. This arrest feature was found to be present when the loads are tensile [6]. The same damage analyses were conducted for panels identical to the flat plates but having a cylindrical shape representing 1/4 of a complete cylinder. No distinct buckling prior to ultimate failure was observed. Tailoring proved to lower the imperfection (local damage) sensitivity of the curved panels when compared to the load–deflection plots of undamaged panels. In fact, in some cases the tailored panel ultimate loads slightly exceeded the theoretical undamaged buckling loads. To the contrary, the ultimate load for the uniform curved panel was found to be much lower than the undamaged buckling load. Relative improvements in ultimate loads ranged from a low of about 40% to a high of about 155%. In the curved panels, damage initiated at the side of the cutout and progressed in a narrow circumferential band (uniform and gently tailored cases) or along an inclined band in the highly tailored cases. Failure occurred after the damage had progressed only part way across the panel, with the extent of the damaged area being larger for the more highly tailored panels. The large difference in the damage initiation and progression between the flat and the curve panels is of some interest. The curved panel damage strongly resembled the previously investigated flat panel damage where outof-plane deformations were prevented [6]. It is believed that the panel curvature provides so much stiffness against these out-of-plane deformations as to make them respond in a way similar to the artificially restrained flat panels.
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The final conclusion drawn from this work, in conjunction with the previous study [6], is that the tailoring concept investigated here can provide excellent improvements in ultimate load capacity in flat and curved with the largest benefits occurring in thin flat panels that are loaded far into the compressive postbuckling regime. Acknowledgements Financial support for this work was provided by NASA, the State of South Carolina, and Clemson University through the EPSCoR grant ‘‘Development and Enhancement of Research Capability for Aircraft Structures and Materials’’. References [1] Biggers SB, Srinivasan S. Compression buckling response of tailored composite plates. AIAA J 1993;31:590–6. [2] Biggers SB, Pageau S. Shear buckling response of tailored composite plates. AIAA J 1994;32:1100–3.
[3] Biggers Jr SB, Browder Jr TM. Buckling-load interaction in tailored composite plates. Compos Eng 1994;4:745–61. [4] Biggers SB, Srinivasan S. Postbuckling response of piece-wise uniform tailored composite plates in compression. J Reinf Plast Compos 1994;13(9):803–21. [5] Baranski AT, Biggers Jr SB. Post-buckling analysis of tailored composite plates with progressive damage. Compos Struct 1999; 46:245–55. [6] Biggers Jr SB, Xie D. Damage progression analysis in tailored laminated plates and shells with a cutout. In: Proceedings of the American Society for Composites, 16th Annual Technical Conference. ASC Paper 136, Blacksburg, VA, September 2001. [7] HKS, Inc. ABAQUS Theory and UserÕs Manual, version 5.8. HKS, Pawtucket, RI. [8] Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I––Analysis. J Compos Mater 1991;25:2–43. [9] Starnes Jr JH, Rouse M. Postbuckling and failure characteristics of selected flat rectangular graphite-epoxy plates loaded in compression. AIAA Paper 81-0543, April 1981. [10] Knight Jr NF, Starnes Jr JH. Postbuckling behavior of axially compressed graphite-epoxy cylindrical panels with circular holes. ASME J Press Vessel Technol 1985;107:394–402. [11] Engelstad SP, Reddy JN, Knight Jr NF. Postbuckling response and failure prediction of graphite-epoxy plates loaded in compression. AIAA J 1992;38:2106–13.