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.

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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:

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

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

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

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

Factor of Safety for Tensile Stress and Shear Stress of Injector Bolts During Hot Fire:

\[ \begin{align*} FOS_{tensile} = \frac{σ_{steel1}}{σ_{hot}} & = \frac{σ_{steel1}}{N_{injector}} = \frac{σ_{steel}N_{injector}πD_{pitch1}^2}{4P_{injector}A_{injector}} \\ & \\ & = \frac{σ_{steel1}N_{injector}π(D_{major1} - (0.6495*t_{s1}))^2}{4P_{injector}A_{injector}} & \\ & \\ & \\ FOS_{shear} = \frac{τ_{steel1}}{τ_{hot}} & ≈ \frac{σ_{steel1}}{\sqrt{3}*τ_{hot}} = \frac{σ_{steel1}N_{injector}πD_{pitch2}^2}{4\sqrt{3}P_{injector}A_{injector}} = \frac{σ_{steel}\sqrt{3}*N_{injector}πD_{pitch2}^2}{12P_{injector}A_{injector}} \\ & \\ & ≈ \frac{σ_{steel1}\sqrt{3}*N_{injector}π(D_{major2} - (0.6495*t_{s2}))^2}{12P_{injector}A_{injector}} \end{align*} \]

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

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