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.
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:
\[ \begin{align*}
& D_{pitch} = D_{major} - (0.6495*t_s) \\
& A_{bolt} = πr^2 = π(\frac{d_{pitch}}{2})^2 = \frac{1}{4}πd_{pitch}^2
\end{align*} \]
Tensile Stress on Each Axial Bolt During Hot Fire:
\[ \begin{align*}
& F_{up} = P_{injector}*A_{injector} \\
& F_{up} + (N*F_{bolt})=0 \\
& \\
&|F_{bolt}| = \frac{P_{injector}*A_{injector}}{N} \\
& \\
& Stress_{Tensile}: σ_{hot} = \frac{|F_{bolt}|}{A_{bolt}} = \frac{P_{injector}*A_{injector}}{N*A_{axial}} \\
& \\
& σ_{steel} = σ_{steel} \\
\end{align*}
\]
Shear Stress on Each Radial Bolt During Hot Fire and Approximation of Shear Strength:
\[ \begin{align*}
& F_{up} = P_{injector}*A_{injector} \\
& F_{up} + (N*F_{bolt})=0 \\
& \\
&|F_{bolt}| = \frac{P_{injector}*A_{injector}}{N} \\
& \\
& Stress_{Shear}: τ_{hot} = \frac{|F_{bolt}|}{A_{bolt}} = \frac{P_{injector}*A_{injector}}{N*A_{radial}} \\
& \\
& τ_{steel} ≈ \frac{σ_{steel}}{\sqrt{3}} \\
\end{align*} \]
Factor of Safety for Tensile Stress and Shear Stress During Hot Fire:
\[ \begin{align*}
& FOS_{tensile} = \frac{σ_{steel}}{σ_{hot}} = \frac{σ_{steel}}{N_{bolt}} = \frac{σ_{steel}N_{bolt}πD_{pitch1}^2}{4P_{injector}A_{injector}}\\
& \\
& \\
& FOS_{shear} = \frac{τ_{steel}}{τ_hot} = \frac{σ_{steel}}{\sqrt{3}*τ_{hot}} = \frac{σ_{steel}N_{bolt}πD_{pitch1}^2}{4\sqrt{3}P_{injector}A_{injector}} = \frac{σ_{steel}N_{bolt}\sqrt{3}πD_{pitch1}^2}{12P_{injector}A_{injector}}\\
\end{align*} \]