Manifold Bolt Calculations

For the injector faceplate to stay fastened to the manifold, the bolts must satisfy a factor of safety for both tensile and shear stress. A net upward pressure acting on the faceplate applied to the area of the injector requires all bolts on the injector manifold -both axial and radial- to withstand this force to maintain structural integrity. These calculations also pertain to the radial bolts keeping the faceplate attached to the combustion chamber.

Desmos Calculator (Injector + Combustion Chamber)

PolarisManifoldBoltCalculations.py

Known Values:

For Injector:

For Combustion Chamber:

Calculations:

The area of the bolt can be determined by using its pitch diameter, which is calculated using its major diameter and the spacing between each thread:

\[ \begin{align*} D_{pitch} & = D_{major} - (0.6495*t_s) \\ A_{bolt} & = πr^2 = π(\frac{d_{pitch}}{2})^2 = \frac{1}{4}πd_{pitch}^2 \end{align*} \]

\[ \begin{align*} & F_{up} = (P_{ox}*A_{ox}) + (P_{fuel}*A_{fuel})\\ & F_{up} + (N*F_{bolt}) = 0 \\ & |F_{bolt}| = \frac{(P_{ox}*A_{ox}) + (P_{fuel}*A_{fuel})}{N_{axial}} \\ & \\ & \\ Stress_{Tensile}: σ_{hot} & = \frac{|F_{bolt}|}{A_{bolts}} = \frac{(P_{ox}*A_{ox}) + (P_{fuel}*A_{fuel})}{N*A_{axial}} \\ & \\ σ_{steel1} & = σ_{steel1} \\ \end{align*} \]

Tensile Stress on Each Axial Bolt During Hot Fire (Outer Bolt Configuration):

\[ \begin{align*} & F_{up} = (P_{ox}*A_{ox}) + (P_{fuel}*A_{fuel}) \\ & F_{up} + (N*F_{bolt}) = 0 \\ & |F_{bolt}| = \frac{(P_{ox}*A_{ox}) + (P_{fuel}*A_{fuel})}{N_{axial}} \\ & \\ & \\ Stress_{Tensile}: σ_{hot} & = \frac{|F_{bolt}|}{A_{bolts}} = \frac{(P_{ox}*A_{ox}) + (P_{fuel}*A_{fuel})}{(N_{inner}*A_{inneraxial})+(N_{outer}*A_{outeraxial})} \\ & \\ σ_{steel1} & = σ_{steel1} \\ \end{align*} \]

Shear Stress on Each Radial Bolt During Hot Fire and Approximation of Shear Strength:

\[ \begin{align*} & F_{up} = (P_{combustion}*A_{combustion}) \\ & F_{up} + (N*F_{bolt})=0 \\ &|F_{bolt}| = \frac{P_{combustion}*A_{combustion}}{N} \\ & \\ & \\ Stress_{Shear}: τ_{hot} & = \frac{|F_{bolt}|}{A_{bolts}} = \frac{P_{combustion}*A_{combustion}}{N*A_{radial}} \\ & \\ σ_{steel2} & = σ_{steel2} \\ & \\ τ_{steel2} & ≈ \frac{σ_{steel2}}{\sqrt{3}} \\ \end{align*} \]

Factor of Safety for Tensile Stress on Injector Bolts During Hot Fire (Standard Configuration):

\[ \begin{align*} FOS_{tensile} = \frac{σ_{steel1}}{σ_{hot}} & = \frac{σ_{steel1}N_{injector}A_{axial}}{(P_{ox}A_{ox}+P_{fuel}A_{fuel})} = \frac{σ_{steel}N_{injector}πD_{pitch1}^2}{4(P_{ox}A_{ox}+P_{fuel}A_{fuel})} \\ & \\ & = \frac{σ_{steel1}N_{injector}π(D_{major1} - (0.6495*t_{s1}))^2}{4(P_{ox}A_{ox}+P_{fuel}A_{fuel})} \end{align*} \]

Factor of Safety for Tensile Stress on Injector Bolts During Hot Fire (Outer Bolt Configuration):

\[ \begin{align*} FOS_{tensile} = \frac{σ_{steel1}}{σ_{hot}} & = \frac{σ_{steel1}((N_{inner}A_{inneraxial})+(N_{outer}A_{outeraxial}))}{(P_{ox}A_{ox}+P_{fuel}A_{fuel})} \\ & = \frac{σ_{steel1}π((N_{inner}(D_{pitch1})^2)+(N_{outer}(D_{pitch2})^2)))}{4(P_{ox}A_{ox}+P_{fuel}A_{fuel})} \\ & \\ FOS_{tensile} & = \frac{σ_{steel1}π((N_{inner}(D_{major1} - (0.6495*t_{s1})^2)+(N_{outer}(D_{major2} - (0.6495*t_{s2})^2)))}{4(P_{ox}A_{ox}+P_{fuel}A_{fuel})} \\ & = 2.1305 \\ \end{align*} \]

Factor of Safety for Shear Stress on Radial Bolts of Combustion Chamber During Hot Fire:

\[ \begin{align*} FOS_{shear} = \frac{τ_{steel2}}{τ_{hot}} & ≈ \frac{σ_{steel2}}{\sqrt{3}*τ_{hot}} = \frac{σ_{steel2}N_{chamber}πD_{pitch3}^2}{4\sqrt{3}P_{chamber}A_{chamber}} = \frac{σ_{steel2}\sqrt{3}*N_{chamber}πD_{pitch3}^2}{12P_{chamber}A_{chamber}} \\ & \\ FOS_{shear} & ≈ \frac{σ_{steel2}\sqrt{3}*N_{chamber}π(D_{major3} - (0.6495*t_{s3}))^2}{12P_{chamber}A_{chamber}} \\ & ≈ 2.6764 \\ \end{align*} \]

Results:

To ensure that the injector manifold bolts can withstand twice the force they are expected to bear, twelve bolts were added to the outer radius. Assuming alloy steel bolts are used and that the expected pressures in both the injector manifold and combustion chamber are accurate:

Tensile Factor of Safety for the Injector Manifold: 2.1305

Shear Factor of Safety for the Radial Bolts on the Combustion Chamber: 2.6764