Thin-Walled Structures Are Highly Susceptible to Buckling

A.5.2.6.1 Global Buckling of Complete Launch Vehicle

Thin-walled structures are highly susceptible to buckling. The design of our launch vehicle is a thin-walled body and therefore, it is necessary to perform a buckling analysis to determine a critical buckling load. In previous sections, local buckling has been discussed. In this section, we consider the overall, global, buckling of the launch vehicle. We determine the critical buckling load of the launch vehicle by using an eigenanalysis.

The stability analyses, such as buckling, occur in two stages. These two stages are the pre-buckle analysis and the buckling analysis. It is important to figure out the geometric stiffness matrix so that in-plane stresses can be determined. We know it is necessary to find the in-plane stress because the presence of in-plane stress causes the onset of buckling. However, the in-plane stresses are usually not known in advance and therefore it is important to allow the degrees of freedom to be such that the in-plane stresses can be evaluated. Then, we solve the eigenvalue problem using the following equation,

臌轾[KE]-l[ K G ] { f} = 0 (A.5.2.6.1.1)

where KE is the elastic stiffness, KG is the geometric stiffness, λ is the eigenvalue, and Φ is the nodal displacements. We complete the buckling analysis by solving for the values of λ that make the system unstable.

In performing the buckling analysis of our launch vehicle, we simplify the vehicle to represent a column buckling problem. When the load on the column is applied through the center of gravity of its cross section, then the load is an axial load. In short columns loaded axially, it is likely that it will fail due to the compression by the axial load before it will fail due to buckling. In long columns loaded axially, the failure will occur as buckling. The critical load for buckling can be stated as the following,

p 2 EI P = (A.5.2.6.1.2) c L2 where Pc is the critical buckling load, E is the modulus of elasticity, I is the second moment of inertia, and L is the length of the launch vehicle. Now, we analyze the above equation using the definition of the second moment of inertia of a hollow cylinder in equation A.5.2.6.1.3.

p 4 4 I=( DO - D I ) (A.5.2.6.1.3) 64

where DO is the outer diameter and DI is the inner diameter. We perform this analysis to allow the affect of the length and radius to be seen explicitly. It is seen that in order to increase the critical buckling load, either the total length has to be decreased or the radius has to be increased.

Now that we understand the necessity of a global buckling analysis and understand what parameters affect the critical buckling load, we construct the algorithm used in the global buckling analysis.

We model the three stage launch vehicle using fifteen elements and sixteen nodes. Each stage contains five elements of equal length. The downfall to this approach is that the element length is not consistent between various sized launch vehicles or stages. The number of elements and nodes are fixed quantities. However, the number of elements used in the analysis is enough to provide reliable values for the first couple modes. In Figure A.5.2.6.1.1, we sketch a rough setup of the global buckling finite element model for the launch vehicle. N16 E15 N15 E14 N14 E13 Material 3 N13 E12 N12 E11 N11 E10 N10 E9 N9 E8 Material 2 N8 E7 N7 E6 N6 E5 N5 E4 N4 E3 Material 1 N3 E2 N2 E1 N1

Fig. A.5.2.6.1.1. Diagram of finite element model for the three stage launch vehicle used for global buckling analysis.

The figure shows that there are fifteen element and sixteen nodes. We designate elements one through five to the first. The second stage has elements six through ten designated to it, and the third stage has elements eleven through fifteen designated to it. The figure illustrates that we assign a different material to each stage to account for the possibility of a different material for every stage plus the different properties such as the cross-section area and the moment of inertia. The figure also demonstrates the boundary conditions and loading conditions applied to the launch vehicle for the global buckling analysis. We fully fix the launch vehicle at node one without any other degrees of freedom released on any of the other nodes in the model. We then apply a compressive axial load to the model at node sixteen. A similar model was created for the two stage launch vehicle except the model uses a total of fourteen elements, which gives seven elements per stage. Similar to the three stage launch vehicle, the two stage launch vehicle also has a different material definition for every stage.

We did not construct the finite element model in any finite element modeling program. Instead, we wrote a program, global_buck.m, using Matlab to generate a text file, or structure data file, that we use in StaDyn, the executable used by QED, to complete the global buckling analysis. We wrote the Matlab code so that a text file could be written which takes in several parameters. The code was written so that the output from main_once.m could be used as the input parameters to the global buckling code. These input parameters include the number of stages in the launch vehicle, the length of each stage, the diameter of each stage, the wall thickness of each stage, the material of each stage, and the compressive axial load. The compressive axial load is the gross lift off weight of the launch vehicle multiplied by the maximum expected gravity loading, which is 6 G’s. After these parameters are input into the code, the code uses the length of each stage to compute the node locations. It then defines the material properties for each stage and computes the area of the cross-section for each stage and the moment of inertia for each stage. After we calculate these values, the values are sent to another code, editfiles2.m, which uses them to write the structure data file. Once editfiles2.m writes the structure data file, it returns to the global_buck.m, which uses a command line along with a command file to run the structure data file for buckling using the StaDyn program. After StaDyn completes its analysis, global_buck.m reads in the output file written by StaDyn with the results of the analysis. The output of interest is the first lambda value. After the code reads in this value, it calculates a maximum G loading that the structure can withstand before buckling as its output. In Figure A.5.2.6.1.2, the process discussed above is written out in a flow chart format so that the code algorithm can be followed more easily. Global Buckling Code

Inputs: No. of stages Stage lengths Stage diameters Wall thickness for stage Material for stage Compressive load

Calculate Applied load

Calculate Calculate 2 3 Node No. of Node locations stages locations

Determine Determine Material Material properties properties

Calculate Calculate stage areas stage areas

Calculate Calculate inertias inertias

Write StaDyn Write StaDyn input deck input deck

Run Run StaDyn StaDyn

no yes Calculate Max. G Λ > 1.25 durability

Output: Output: Launch Vehicle Maximum failed withstandable G’s

Fig. A.5.2.6.1.2. Flow chart of algorithm used for global buckling analysis We ran global_buck.m for the final design of the 200 g payload, 1 kg payload, and 5 kg payload launch vehicles. We summarize the results from our analysis in Table A.5.2.6.1.1.

Table A.5.2.6.1.1 Maximum G’s the Launch Vehicle can Withstand. Launch Vehicle Max G durability Units 200 g payload 141.8 G’s 1 kg payload 246.7 G’s 5 kg payload 53.1 G’s Footnotes:

Dr. James Doyle, a professor of Aeronautics and Astronautics Engineering at Purdue University, suggested using a knockdown factor of 0.60 to account for reductions in the strength due to manufacturing and imperfections of the material. The values listed above use a knockdown factor of 0.50 to account for the topics brought to out attention and to allow some error for using a simplified column buckling approach. In addition to the knockdown factor, the factor of safety equal to 1.25 was also used in reporting the results listed above. Although we do not know if the knockdown factors we applied are enough to allow for error using the simplified approach, the maximum withstandable gravity loading predicted by the analysis are much higher than the 6 G’s that we expect the launch vehicle to experience. Therefore, we conclude that global buckling should not present any problems for our launch vehicle.

When we first looked at the results from our analysis, we were interested to see the difference in each launch vehicle that would produce the differences in the maximum gravity loading durability. In the tables below, the length and diameter of each stage are displayed to help grasp the results of the analysis. Table A.5.2.6.1.2 200 g Payload Launch Vehicle Length and Diameter of Each Stage. Stage Length Diameter Units First 4.94 0.83 m Second 1.77 0.23 m Third 0.99 0.06 m Footnotes:

Table A.5.2.6.1.3 1 kg Payload Launch Vehicle Length and Diameter of Each Stage. Stage Length Diameter Units First 4.16 0.66 m Second 1.48 0.19 m Third 1.06 0.07 m Footnotes:

Table A.5.2.6.1.4 5 kg Payload Launch Vehicle Length and Diameter of Each Stage. Stage Length Diameter Units First 7.07 1.24 m Second 2.19 0.30 m Third 1.00 0.06 m Footnotes:

Table A.5.2.6.1.5 Liftoff Mass for All Three Launch Vehicles. Launch Vehicle Liftoff Mass Units 200 g payload 2890.33 kg 1 kg payload 1819.86 kg 5 kg payload 6372.71 kg Footnotes:

We look at these results and recall equation A.5.2.6.1.3 to see what produced the differences in the maximum gravity loading durability. We compare the 200 g payload launch vehicle and the 1 kg payload launch vehicle and see that the 200g payload launch vehicle is larger in its geometry and is more massive than the 1 kg payload launch vehicle by 1.6 times. The fact that the vehicle geometry increased in a smaller proportion than the mass in the comparison of the two launch vehicles, results in the higher gravity loading capability of the 1 kg payload launch vehicle over the 200 g payload launch vehicle. Now we compare the 1 kg payload launch vehicle and the 5 kg payload launch vehicle. As in the case before, the ratio between the geometry of the two launch vehicles is less than half the ratio between the masses of them. This, again, results in the higher gravity loading capability of the 1 kg payload launch vehicle over the 5 kg payload launch vehicle.

References:

James F. Doyle. Professor of Aeronautics and Astronautics Engineering. Purdue University.

Doyle, James F., Guided Explorations on the Mechanics of Solids & Structures: strategies for learning and understanding. Purdue University, West Lafayette, IN. August 2007.

Doyle, James F., Structural Dynamics and Stability: a modern course of analysis and applications. Purdue University, West Lafayette, IN. August 2007. ikayex Software Tools. QED: Static, Dynamic, Stability, and Nonlinear Analysis of Solids and Structures. Lafayette, IN. August 2007.