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A.5.2.2 Inter-Stage Skirts
A. 5.2.2.3 Effect of Thickness on Buckling
Buckling analysis of a long and slender object, such as our launch vehicle, is very important in determining its structural stability. Generally, buckling occurs with loads below the values of ultimate and compressive stress.1 Therefore, we see that buckling governs the design of these objects, above strength considerations.
Although this value governs the design of the launch vehicle, exact solutions are extremely difficult and sometimes impossible to determine.1 In most instances, estimations, assumptions, and numerical methods are necessary to determine the critical buckling values. These methods are very useful to our analysis because trying to develop more exact solutions may not be helpful.
Many different methods have been developed for this application. For our analysis, we will explore three of these methods, all of which were derived from Von Karman’s equations.1
The first method1for analyzing buckling begins with the general equation below.
(A.5.2.2.3.1)
where D is the bending rigidity (N/m), w is the radial deflection (m), x is the axial coordinate (m), N is the uniform axial compressive force (N), E is Young’s modulus (Pa), h is thickness of the shell (m), and R is the radius (m).
The exact solution from this general equation is then given by Eq. (A.5.2.2.3.2).
(A.5.2.2.3.2)
Where C is a constant of integration.
Simplifying, the critical force can be found as
(A.5.2.2.3.3)
whereμ is Poisson’s ratio for the material.
The second method2for analysis of buckling assumes an unpressurized, long cylindrical shell. The critical stress value is found via Eq. (A.5.2.2.3.4).
(A.5.2.2.3.4)
Where Cc is approximately equal to 0.6, E is Young’s modulus (Pa), t is the thickness of the shell (m), R is the radius of the cylinder (m), and γ is representative of the R/t ratio.
The third and final method3that we incorporate in our analysis of buckling is considered the classic solution for axially compressed cylinders. It simplifies to the following form
(A.5.2.2.3.5)
Where E is Young’s modulus (Pa), h is the thickness of the shell (m), and ais the length of the cylinder (m).
In our design, the length is dictated by propulsion systems, tanks, and other equipment. To ensure the stability of the launch vehicle, component thickness is varied to reach the necessary requirements. The trends seen from the three different methods are also very different in value and shape.
Fig. 5.2.2.3.1: Method 1 Trends for Critical Pressure
(Molly Kane)
Method 1 shows an exponential rise in thickness for increased critical pressure.
Fig. 5.2.2.3.2: Method 2 Trends for Critical Pressure
(Molly Kane)
Method 2, however, shows a linear relationship, as does Method 3 below.
Fig. 5.2.2.3.3: Method 3 Trends for Critical Pressure
(Molly Kane)
Overall, the first method1 gives a low-end calculation to ensure the stability of our launch vehicle. However, through this research it is seen that a more in-depth analysis of the launch vehicle must be completed with piecewise steps and including the pressurized tank analysis. This essentially divides the launch vehicle into many smaller elements and allows thicknesses to be changed for each part, rather than for the entire structure.
References
1Wang, C.Y., Wang, C.M., Reddy, J.N., Exact Solutions for Buckling of Structural Members, CRC Press, Boca Raton, FL, 2005.
2Brush, D.O., Almroth, B.O., Buckling of Bars, Plates, and Shells, McGraw Hill, 1975, pgs. 161-165.
3Baker, E.H., Kovalevsky, L., Rish, F.L., Structural Analysis of Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981, pgs. 229-240.
Author: Molly Kane