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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.


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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:

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


Tensile Stress on Each Axial Bolt:


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


Shear Stress on Each Radial Bolt and Approximation of Shear Strength:

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


Factor of Safety for Tensile and Shear Stress:

\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 = \frac{|F_{bolt}|}{A_{bolt}} = \frac{P_{injector}*A_{injector}}{N*A_{bolt}} \\ & \\ & τ_{steel} ≈ \frac{σ_{steel}}{\sqrt{3}} \\ \end{align*}


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