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Figures 5-7A, 5-7B and 5-7C illustrate the results of the ITPS panel Design 1. During the reentry phase (0 to 2175 seconds) in-plane loads are applied on the panel and after landing (after 2175 seconds) pressure load is applied on the top face sheet (see Section 5.1 for details). In all the three figures, the temperature variation of top face sheet, bottom face sheet and the temperature difference between the two face sheets has been illustrated and the values correspond to the axis on the left. Absolute value of the temperature difference has been plotted.
The increase in temperature difference after landing is a result of plotting this absolute value. A little while after landing the bottom face sheet temperature becomes higher than the top face sheet temperature and the difference increases till the bottom face sheet reaches its peak temperature and drops down after that. tmax∆T is at 650 seconds and tmaxBFT is at 3025 seconds.
From Figure 5-7A, it can be observed that the minima for top face sheet and web buckling is at 450 seconds. This time is different from tmax∆T (650 seconds) and is the point where the temperature of the bottom face sheet is still close to its initial temperature of 295 K. Another interesting point is the second minima in the web buckling curve which occurs immediately after landing. The reason for this dip in the curve is due to the application of pressure load on the top face sheet after landing. However, even though this pressure load is unchanged after landing, it can be observed that the web buckling eigenvalue increases. The reason for this is that the ITPS panel begins to cool down after landing and the average temperature of the web also decreases.
As the stiffness of the webs (Young’s modulus) increases with decrease in temperature, the buckling eigenvalue of the web also increases.
Figure 5-7B shows the variation of top face sheet deflection with reentry time. Absolute value of the deflection has been plotted in this figure. The first maxima occurs at 650 seconds corresponding to the time when temperature difference between the face sheets is maximum.
This temperature difference causes the top face sheet to bow “convex-up”. After landing the pressure load causes the top face sheet to deflect downward (concave-up). Since, the plot is for absolute value of deflection, the curve rises again after landing and the second maxima occurs at around 3,025 seconds, which corresponds to the time when bottom face sheet reaches its peak temperature value. The increase in deflection during the early phase of cool-down (after landing) can be attributed to the increase in bottom face sheet temperature, which decreases the stiffness of the bottom face sheet material.
Figure 5-7C shows the variation of stresses with respect to reentry time. The stresses peak at either 450 or 650 seconds and then again at around 3,025 seconds.
The reentry behavior of the ITPS panel corresponding to Design 2 is shown in Figure 5-8.
The buckling behavior, Figure 5-8A, during reentry phase is similar to that of Design 1.
However, the second minima that occurs in Design 1 after landing is not present in this case. The difference is due to the large number of webs (number of unit cells = 10) in Design 2. The pressure load is distributed among the webs and so this buckling mode does not surface here.
Deflection values, Figure 5-8B, are higher for Design 2 because the panel is longer in this case.
The larger panel length also leads to higher stresses, Figure 5-8C, in Design 2.
600 1.5 400 1.0 200 0.5
Figure 5-7. ITPS panel behavior with respect to reentry time for Design 1. In all three figures, the temperature variation of top face sheet, bottom face sheet and the temperature difference between the two have been plotted. A) Buckling eigenvalues. B) Top face sheet deflection (maximum allowable deflection is 6 mm). C) Stresses in top face sheet, bottom face sheet and webs. Allowable stress in top face sheet and webs is 620 MPa and in bottom face sheet is 290 MPa.
The important conclusions of this study are the identification of critical reentry times for buckling and stress analyses. The critical times for buckling analyses are • 450 seconds (when the temperature of the bottom face sheet is close to its initial temperature), and • 2225 seconds (just after landing and at the beginning of the application of pressure load).
The critical times for stress and deflection analyses are:
• when the temperature difference between the face sheets is maximum, and
• when the bottom face sheet temperature is maximum.
3.0 2.0 1.0
Figure 5-8. ITPS panel behavior with respect to reentry time for Design 2. In all three figures, the temperature variation of top face sheet, bottom face sheet and the temperature difference between the two have been plotted. A) Buckling eigenvalues. B) Top face sheet deflection (maximum allowable deflection is 6 mm). C) Stresses in top face sheet, bottom face sheet and webs. Allowable stress in top face sheet and webs is 620 MPa and in bottom face sheet is 290 MPa.
Due to the boundary conditions imposed on the panel edge, bottom face sheet stresses for both designs are close to the limit stress of 340 MPa for Beryllium. The reasons for the high stresses have been discussed in the previous section. It has been observed from the optimization procedure that increasing the factor of safety to more than 1.2 does not yield any optimized designs as the bottom face sheet stress constraints cannot be satisfied. As many of the input parameters assumed for the study are approximate in nature, there is very little room to tweak these values to study their effect on the weight of the ITPS. Therefore, there is a need to relax the boundary conditions in order to reduce the stresses developed in the bottom face sheet. New boundary conditions and the subsequent optimized designs are described in the following section.
5.2.4 Optimized Designs with Changed Boundary Conditions In order to reduce the stresses in the bottom face sheet the boundary constraints were relaxed. Simply supported boundary conditions, which were previously imposed on both the panel edges, were imposed only on Edge A of Figure 3-9, and Edge B was left free. All other loads and boundary conditions were same as the previous corrugated-core design. The constraints imposed in the optimization problem were also same except for the factor of safety on stresses, which was increased to 1.5, which is the usual value used in structural design. The accuracy of response surface approximations was similar to those listed in Tables 5-3 to 5-5.
The new optimized designs are shown in Table 5-7. All four designs have the same active
• Bottom face sheet temperature
• Top face sheet deflection
• Web buckling eigenvalue The deflection constraint becomes active because the panels in the new designs are longer.
Due to the new boundary conditions, none of the stress constraints are active in spite of the increase in the factor of safety. Web buckling was observed to be caused by pressure load on the top face sheet (which is applied after landing).
Comparison of values predicted by response surface approximations and the finite element analysis are presented in Table 5-8 for Designs I and IV. Values are close to one another indicating that the response surface approximations are sufficiently accurate. As expected buckling eigenvalues show the largest error. However, the conservative value of 1.25 on buckling eigenvalue constraints is sufficient to take this inaccuracy into account.
Truss core structures were explored for the ITPS applications as they were perceived to provide a lighter structure when compared to the corrugated-core structures. One of the major advantages of truss-core structures is that the trusses provide a much smaller path for heat flow and this would lead to a better insulated ITPS. Trusses are structurally superior to plates (webs in corrugated-cores) in carrying compressive loads that would arise as a result of pressure loads on the top face sheet. They are also better suited to carry bending loads when compared to plates.
Unlike the corrugated-core webs, trusses do not tend to exert large forces on the face sheets and they also do not offer large restraining forces on the face sheets that would lead to higher stresses.
In spite of the seemingly overwhelming advantages of truss-cores when compared to corrugated-cores, it was found that the trusses are too weak to withstand the large bending forces created in them by the severe thermal gradient in the ITPS panels. Trusses also produce large stress concentrations in the face sheets at the points of attachments. In this section, some of the work done on the truss-core structures is presented for the advantage of researchers who want to pursue this line of design for ITPS applications.
A typical FE model with truss-cores is shown in Figure 5-9A. As in the case of corrugatedcores, the face sheets were modeled with the shell element. The trusses were modeled with the 3node (quadratic) beam element (ABAQUS element B32 and B32OS). All loads and boundary conditions imposed were similar to the corrugated-core models. ABAQUS has a provision to specify the beam cross-section. Various cross-sections like pipe-, box-, solid-rectangular- and solid-circular-sections were used. It was found that the I-section produced the minimum stresses of all cross-sections, when oriented in a proper direction.
Irrespective of the beam cross-section, stress singularity at the beam-plate attachment points was found to be big problem in the FE modeling of truss-core panels. In Figure 5-9A, some of the attachment points are pointed to by the arrows where the stresses produced are the highest. As expected, the stresses in the face sheets and the webs are highest in regions closest to the panel edges. At first look, the high stresses at the attachment points appeared to be a case of stress concentration arising out of a modeling error. However, on closer examination it was found that there exists a singularity in the face sheets due to the point loads and moments exerted by the beams on the face sheets. In Figures 5-9B-5-9D a patch of the top face sheet near the corner-most junction point is shown with increasing mesh refinement in the model. The stresses at the junction node are listed below each patch. The stresses in the top face sheet increase with mesh refinement almost doubling in magnitude whenever the element size is reduced by half.
The stresses in the beams, however, do not show such a drastic change. In fact, the stresses in the beams decrease with mesh refinement. It is quite clear that the stresses in this model are not true stresses as their magnitude is heavily influenced by the mesh refinement. Further, the mesh refinement also alters the force and moment interactions between the plates and the beams. As the mesh is refined, the patch around the junction points becomes more compliant and thus a decrease in the beam stresses can be observed.
In order to further understand the shell-beam interactions, an FE experiment was conducted, where in one unit cell of the truss-core model was modeled with beam elements and shell elements, as shown in Figure 5-10. The bottom face sheet was constrained on the edges and an arbitrary edge load was applied on the top face sheet. The stresses for increasing mesh refinement are listed in Table 5-9. As expected, the beam model exhibited stress singularity and the top face sheet stresses increased with mesh refinement. The shell model, however, showed a slight increase in stresses and it can be interpreted that the stresses in this case are stress concentrations arising as a result of modeling error. Singularities do not affect the stresses in the beam. In both cases stresses in truss tend to converge with mesh refinement. However, the stresses are much lower for the beam model. It can be concluded that the force and moments exerted by the face sheet on the truss are not accurate in the case of beam model. This is an additional inaccuracy in the beam model, apart from the stress singularities.
Figure 5-9. Stress singularity at beam-plate junction points. A) Full FE model showing stress singularity points at various junction points. B)–D) A patch of the top face sheet at the junction point closest to the panel corner shown for increasing mesh refinements.
The von Mises stresses (listed below the patches) in the top face sheet at the junction node are listed as ‘TFS’ and the web stresses are listed as ‘Beam’.
In order to study the actual stresses in the truss-core ITPS model, the entire panel was modeled with shell elements as shown in Figure 5-11. Such a model is difficult to implement with the ITPS Optimizer because each model analysis is very time consuming. The stresses in the trusses, Figure 5-11C, were found to be over 1 GPa. Increasing the stiffness of the trusses by increasing the wall thickness of the trusses leads to higher stress concentrations in the face sheets.
Figure 5-10. FE experiment to compare beam model to a more realistic shell model. A) The trusses are modeled with the beam element. B) Trusses are modeled as the shell element. In both cases the truss section is a pipe cross-section. The arrows show the load. The bottom face sheet is completely restrained on the edges.
The results presented in this section provide insight into the behavior of truss-core structures subjected to typical ITPS loads. Feasible light weight designs could not be obtained with the truss-cores. Therefore, this line of design improvement was not considered further in the design process. It should be noted, however, that a suitable geometric rearrangement of the trusscores could possibly lead to better designs and should be explored in future.
Figure 5-11. Truss-core model with only shell elements. A) Entire FE model is shown with deformation due to temperature load. B) Close-up of the corner junction points where stresses are maximum. C) Only the trusses are shown to illustrate the stress distribution.