Section A.5.2.1.6.2 Effect of Pressure on Buckling and Bending Strength pg.1

A.5.2.1.6.2Effect of Pressure on Buckling and Bending Strength

A pressurized vessel is stronger against buckling and bending than an unpressurized vessel. This is because internal pressure stiffens the structure, provides resistance to axial loads, and minimizes stress concentrations due to local geometric imperfections. In our analysis of the propellant tanks, we take advantage of this phenomenon to optimize our structure and to avoid having to add unnecessary internal supports.

For our analysis, we utilize curves based on empirical test data as presented in Bruhn Chapter C81. Where possible, 90% probability curves with a confidence level of 95% are chosen, which meets the required 90% probability requirements against catastrophic failure specified in MDR-003.

A.5.2.1.6.2.1Tank Buckling

Tank buckling is first modeled using Baker method 32, which gives us the nominal buckling strength of an unpressurized tank.

/ (A.5.2.1.4.3a)

(See Section A.5.2.1.4.1 for more details)

We then employ empirical design curves from Bruhn Fig C8.111 to calculate the proportional increase in buckling strength due to pressurizing the tank.

/ (A.5.2.1.4.3d)

wherePcr is the non-dimensionalized increase in critical buckling strength (see Section A.5.2.1.6.2) and Pcr_press is the critical buckling stress of a pressurized tank, (Pa).

The non-dimensional increase in axial buckling strength Pcr is calculated by plotting data points from Bruhn Fig C8.11 into Mathcad and using the linfit function to curvefit a function to describe the relation in Matlab.

Fig. A.5.2.1.6.2.1Non-dimensional increase in axial buckling strength versus tank internal pressure and dimensions showing function fit curve

Using Mathcad, we obtain a high-correlation function fit of

/ (A.5.2.1.6.2.1)

where Pcr is the non-dimensionalized increase in axial buckling strength, Ptank is the internal pressure in the tank (Pa), E is the Young’s modulus of the material (Pa), Dtank is the tank diameter (m) and t is the tank wall thickness (m).

The range of preliminary tank dimensions and pressures gives us between 15% - 20% improvement in axial buckling stress for a pressurized cylindrical tank.

A.5.2.1.6.2.1Tank Bending

Tank bending allowables are determined from Bruhn Fig C8.13a1. The values from Bruhn are determined from experimental results, and we plot data points and graph them in Microsoft Excel with a log curvefit to derive an expression for allowable bending stress for a given L/D ratio.

Fig. A.5.2.1.6.2.2 Allowable bending strength for an unpressurized cylindrical tank

From Figure A.5.2.1.6.2.2, we obtain the following curvefits for unpressurized tank bending allowable values:

For L/Dtank = 2,

/ (A.5.2.1.6.2.2a)

For L/Dtank = 4,

/ (A.5.2.1.6.2.2b)

For L/Dtank = 8,

/ (A.5.2.1.6.2.2c)

where L is the tank length in m, Dtankis the tank diameter in m, Fbcr is the critical bending stress in Pa, E is Young’s modulus in Pa, t is the tank wall thickness in m.

For L/D values between two lines, we choose the lower line (ie higher L/D ratio) for conservativeness.

We then employ Bruhn Figure C8.14 to determine the increase in bending strength of a pressurized tank. Data points from Bruhn are plotted into Microsoft Excel and a log curvefit is applied.

Fig.A.5.2.1.6.2.3 Non-dimensional increase in pressurized tank bending stress

From Figure A.5.2.1.6.2.3, we obtain an equation for the non-dimensional increase in bending strength for a pressurized tank:

/ (A.5.2.1.6.2.3)

And the final bending allowable

/ (A.5.2.1.6.2.4)

where Fbcrpress is the bending stress allowable for a pressurized tank in Pa, Fbcr is the bending stress allowable for an unpressurised tank in Pa, Cb is the non-dimensionalized increase in bending strength, Ptank is the internal pressure in the tank in Pa, E is the Young’s modulus of the material in Pa, Dtank is the tank diameter in m and t is the tank wall thickness in m.

For the preliminary design candidates, this resulted in a bending stress increase of 30-40% over an unpressurized tank.

References

1 Bruhn, E.F., Analysis and Design of Flight Vehicle Structures, Jacobs Publishing, 1973, Chapter C8 pgs. 347-353

2Baker, E.H., Kovalevsky, L., Rish, F.L., Structural Analysis of Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981, pgs. 229-240

Author: Chii Jyh Hiu