«DESIGN OF AN INTEGRAL THERMAL PROTECTION SYSTEM FOR FUTURE SPACE VEHICLES By SATISH KUMAR BAPANAPALLI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL ...»
Thus there are 7 design variables in all. The product of n and 2p gives the length of the whole ITPS panel, L. It makes more sense to have L as a design variable rather than 2p. Therefore, the sixth design variable 2p is replaced by L. The range of each variable is listed in Table 5-2. These ranges were chosen based on many optimization studies. The ranges were wider for initial studies. They were narrowed down subsequently, as the optimization studies provided insight into the behavior of the ITPS panel. For example, initial range for the top face sheet thickness, tT, was 0.5 to 5 mm. However, optimization studies with different loads, boundary conditions and materials indicated that typical values of tT are closer to 1.0 mm. Thus the range was narrowed down and this also helped in improving the accuracy of the RS approximations.
Titanium alloy, Ti-6Al-4V, is used for top face sheet and web and a beryllium alloy is used for the bottom face sheet. All the input parameters, loads and boundary conditions have been discussed in the previous section. The in-plane loads (discussed in previous section) are applied on the bottom face sheet at the panel edge, because these loads are transmitted by the stringers and frames to the bottom face sheet.
Table 5-2. Ranges of the 7 design variables for corrugated-core ITPS panels tT 1–2 mm tB 2–8 mm tW 1–2 mm θ 80°–100° h 80–120 mm L 450–900 mm n 4–10 At this point, it is necessary to identify the “critical reentry times”, which means the reentry times at which the loads on the ITPS panel cause failure in the form of buckling, excessive stress or deflection. One of the critical reentry times is that at which the temperature difference between the top face sheet and the bottom face sheet is maximum. At this time, the thermal stresses can be expected to be at the maximum. This happens at the initial portion of the reentry phase when the top face sheet is close to its peak temperature and the bottom face sheet is still close to its initial temperature. At this point there are in-plane loads applied on the ITPS panel as well. The combined effect of thermal loads and in-plane mechanical loads is of interest at this point. This critical reentry time will, henceforth, be referred to as ‘tmax∆T’.
Another critical time of interest occurs after landing and when the bottom face sheet reaches its peak temperature value. This is the time when the stresses in the bottom face sheet might be at the maximum. Further, at this time external pressure loads are imposed on the ITPS panel. The combined effects of temperature and pressure loads are of interest at this point. This time will, henceforth, be referred to as ‘tmaxBFT’.
The ITPS Optimizer first carries out the heat transfer analysis and obtains the temperature distribution at tmax∆T and tmaxBFT. These temperature distributions along with the respective mechanical loads are imposed on the structure to carry out the buckling and stress.
5.2.1 Accuracy of Response Surface Approximations The minimum number of responses for obtaining response surface approximations with 7 design variables is 36. For the corrugated-core design, FE analyses were carried out at 250 design points obtained by optimized Latin-Hypercube Sampling Technique (see Section 4.2).
The accuracy of temperature and deflection response surface approximations is presented in Table 5-3, buckling in Table 5-4 and stresses in Table 5-5. The column header ‘eRMS %’ stands for percentage of the root mean square error when compared to the average response value.
Column header ‘ePRESS %’ stands for percentage of PRESS error when compared to the average response value. PRESS is an acronym for Predicted Error Sum of Squares and is a better indicator of the accuracy of the response surface approximations.
The procedure for calculating the PRESS error is as follows. Consider that there are N response values from which the response surface approximation is calculated. First calculate the response surface approximation R1 using responses 2 to (N - 1) by leaving out the first response.
Now determine the error (e1) of R1 by substituting the variable values corresponding to response 1 into this response surface approximation. Similarly calculate R2 to RN and the corresponding errors e2 to eN. The root mean square of the errors e1 to eN is the PRESS error for the response surface approximation.
The buckling response surface approximations are calculated only for tmax∆T because it was found that the eigenvalues for tmaxBFT are always higher that those at tmax∆T. Therefore, eigenvalues at tmaxBFT will not be among the active constraints for the ITPS design. Further, there is no bottom face sheet buckling case because the number of responses for this case is not sufficient to obtain the response surface approximation. The bottom face sheet buckling will not be an active constraint in the design process.
The large error in the top face sheet buckling response surface approximation is due to availability of less number of responses. Among the 250 design points, only 190 points yielded top face sheet buckling eigenvalues. That is, in the remaining design points, the top face sheet buckling did not figure in the first 15 eigenvalues (see Section 4.1.2 for details). Compared to the top face sheet, 234 out of 250 design points yielded web buckling eigenvalues and that is the reason for the higher accuracy of the response surface approximation corresponding to web buckling.
The last row in Table 5-4 referred to as ‘Combined buckling’ is the response surface approximation obtained by using the lowest eigenvalues at each design point, irrespective of top face sheet buckling or web buckling. Note that for case of “combined buckling” 250 out of 250 responses are available for the calculation of the response surface approximation. Yet, it can be observed that this response surface approximation has a higher PRESS error than that compared to the web buckling case. Thus, it is a very good practice to separately obtain response surface approximations for top face sheet and web buckling in spite of the large error in top face sheet buckling case. Another advantage of splitting the buckling response surface approximation is that the active constraints can be separately identified in the optimization procedure, which is not possible in the case of ‘Combined buckling’ response surface approximation.
The accuracy of the response surface approximations corresponding to stresses is shown in Table 5-5. The FE analysis of the corrugated-core ITPS panel showed that there are stress concentrations at the panel edges at the junction between the webs and the face sheets as shown in Figure 5-4. These stress concentrations are an artifact of the FE model, that is, they can be classified as modeling errors. In actual corrugated-core construction, due to the manner in which the webs are attached to the face sheets there will be no such stress concentrations. These stresses are not “true stresses”. Therefore, the stresses at this panel edge (shown by arrows in Figure 5-4) are separated from the stresses in the rest of the panel. The first 3 rows in Table 5-5 show the stresses in the rest of the panel, while the ones in the last 3 rows are the edges stresses.
One of the major reasons for the large errors in some of the response surface approximations is due to the inclusion of the design variable n (the number of unit cells in the panel. Unlike the other 6 variables that influence only the sizing of the panel, the variable n changes the entire geometry and stress transfer mechanisms in the ITPS panel. This leads to a large variation in the eigenvalues and stresses and decreases the accuracy of the response surface approximations. In spite of the large PRESS errors, the response surface approximations can be expected to be qualitatively reasonable and lead to the design close to the real design. It is worthwhile to point out the fact that the percentage RMS error is smaller than 6% in all cases.
At this juncture, it is also appropriate to highlight the fact that the objective of this research is not to accurately design an ITPS panel, but only to investigate the feasibility of such a structure for ITPS applications. An actual ITPS would involve a lot more factors than those considered in this research. Some of the other factors are manufacturing aspects such as joining of dissimilar materials and crashworthiness of the structure. In this light, the design process with the above listed response surface approximations will shed valuable light into the issues related to the design of corrugated-core panels for ITPS applications.
5.2.2 Optimized Corrugated-Core Panel Designs The response surface approximations, discussed in the previous section, are used for the design optimization of the corrugated-core panels. The optimization procedure has been described in Section 4.3.
The following constraints were imposed on the optimization problem
1. Peak bottom face sheet temperature ≤ 200 °C
2. Buckling Eigen Value ≥ 1.25
3. Top face sheet deflection ≤ 6.0 mm
4. Factor of Safety for stresses = 1.2 The temperature and deflection constraints were similar to those imposed on the ARMOR TPS . Although, the limit for minimum buckling eigenvalue is 1.0, a value of 1.25 was chosen to account for the inaccuracy of the buckling response surface approximations. For stresses, typical factor of safety is 1.5. However, a value higher than 1.2 did not produce feasible results in the optimization procedure due to high stresses in bottom face sheet.
Figure 5-4. A typical FE contour plot illustrating the stresses at the panel edges at the junction between the face sheets and the webs. The arrows point to the areas of stress concentrations.
The optimized designs are listed in Table 5-6. The first column of the table lists the intervals within which the respective variables vary. There are 4 designs listed in the table and each is obtained by imposing an additional constraint on the number of unit cells, n. For example, for Design 1 the number of unit cells is made equal to 4, for Design 2 the number of unit cells is made equal to 10 and so on.
Design 1 is the lightest of all, while Design 2 has the advantage of being the longest of all panels and also has the shortest panel height. Design 3 and 4 are heavier and do not have any apparent advantages over the other designs. Longer panels require less number of stringers and frames in the space vehicle. This could help reduce the overall weight of the vehicle outer structure.
The active constraints for Design 1 are stresses in the bottom face sheet, web buckling and top face sheet buckling. For Design 2, apart from these constraints the peak bottom face sheet temperature constraint also become active. In order to better understand the behavior of the panels, it is necessary to understand the buckling modes and stress distribution in these panels.
The web and top face sheet buckling modes for Design 1 and Design 2 are shown in Figure 5-5. As mentioned in Chapter 3, only one quarter of each panel is modeled in the FE model by taking advantage of symmetry. In Figure 5-5A, the panel edges and the symmetry edges are labelled. All the other figures are shown with the same orientation. In the web buckling case, the buckling modes in both designs are similar. These buckling modes happen at tmax∆T when the temperature difference between the bottom face sheet and top face sheet is maximum. The top face sheet is close to its peak temperature and tends to expand. However, the bottom face sheet, which is at much lower temperature, resists this expansion. As the webs transmit the forces between the two plates, they experience large flexural loads. The web that buckles is the farthest from the center of the panel and experiences the highest load. Boundary conditions imposed on the bottom face sheet edges also provide resistance to deformation and this effect is experienced by the web closest to the panel edge.
Figure 5-5. Buckling modes for optimized designs. A) Web buckling for Design 1. B) Top face sheet buckling for Design 1. C) Web buckling for Design 2. D) Top face sheet buckling for Design 2.
At tmax∆T, the top of the webs are at a much higher temperature when compared to the bottom. Thus the web deformation would be as illustrated in Figure 5-6. This deformation is resisted by the face sheets and the resistance by the bottom face sheet is much higher because it is the thickest section of the panel and is also the most constrained member of the panel. This leads to very high stresses in the bottom face sheet, particularly at the web-bottom face sheet junction closest to the panel edge. The stress distribution in the bottom face sheet is shown in Figure 5-7A. Stresses are highest at the region pointed to by the arrow. This region corresponds to the centerline (dash-dot line) in Figure 5-6 where the vertical displacement is the highest in the web. The stress distribution in the top face sheet and the webs are also shown in Figure 5-7. As mentioned earlier, stresses are highest in the sections closest to the panel edges.
C Figure 5-7. Von Mises stress distribution in the ITPS panel, Design 1, tmax∆T. A) Bottom face sheet stresses. B) Web stresses. C) Top face sheet stresses.
5.2.3 Buckling Eigen Values, Deflections and Stresses at Different Reentry Times The two reentry times at which the buckling and stress analyses were carried out are tmax∆T, when the temperature gradient in the structure is most severe, and tmaxBFT, when the bottom face sheet reaches its peak temperature. These two times were chosen on the basis of intuition of where the critical stress and buckling cases could occur. In order to determine whether these two reentry times are the only critical cases, a study was conducted to understand the variation of smallest buckling eigenvalues, maximum deflections and maximum stresses with respect to reentry time. These studies were conducted on the ITPS panels with dimensions of Designs 1 and 2.