Determine the adequacy of a rectangular cross-sectional header with a design pressure of 150 psi and made from a seamless forging with an allowable stress of 12,500 psi. The header inside dimensions are 14 in. by 7.25 in. with a constant thickness of 1 in. All openings are reinforced.

Question

Accepted Answer

1 . Calculate the moment of inertia:

[latex]I=\frac{d^{3}}{12}=\frac{(1)^{3}}{12}=0.0833[/latex]

2 . Calculate bending moment at corner [latex]M_{Q}[/latex] using Eq. 17.18:

[latex]M_{Q}=\frac{P}{12}\left(\frac{h^{3} / I_{2}+H^{3} / I_{1}}{h / I_{2}+H / I_{1}}\right)[/latex] (17.18)

[latex]M_{Q}=\frac{150}{12}\left(\frac{(14)^{3} / 0.0833+(7.25)^{3} / 0.0833}{14 / 0.0833+7.25 / 0.0833}\right)=1838.28[/latex]

3 . Calculate bending moment at midpoint of long-side [latex]M_{M}[/latex] using Eq. 17.19:

[latex]M_{M}=M_{Q}-\frac{P h^{2}}{8}[/latex] (17.19)

[latex]M_{M}=M_{Q}-\frac{P h^{2}}{8}=1838.28-\frac{(150)(14)^{2}}{8}=-1836.72[/latex]

4 . Calculate bending moment at midpoint of short-side [latex]M_{N}[/latex] using Eq. 17.20 :

[latex]M_{N}=M_{Q}-\frac{P H^{2}}{8}[/latex] (17.20)

[latex]M_{N}=M_{Q}-\frac{P H^{2}}{8}=1838.28-\frac{(150)(7.25)^{2}}{8}=852.73[/latex]

5 . Calculate bending stress at corner of long-side:

[latex]\left(S_{b}\right)_{Q M}=\frac{(1838.28)(0.5)}{(0.0833)}=11,030 psi[/latex]

6 . Calculate bending stress at midpoint of long-side:

[latex]\left(S_{b}\right)_{M}=\frac{(1836.72)(0.5)}{(0.0833)}=11,020 psi[/latex]

7 . Calculate bending stress at corner of short-side:

[latex]\left(S_{b}\right)_{Q N}=11,030 psi[/latex]

8 . Calculate bending moment at midpoint of short-side:

[latex]\left(S_{b}\right)_{N}=\frac{(852.73)(0.5)}{(0.0833)}=5120 psi[/latex]

9 . Calculate membrane stress on long-side:

[latex]\left(S_{m}\right)_{M}=\frac{P H}{2 t}=\frac{(150)(7.25)}{2(1)}=540 psi[/latex]

10 . Calculate membrane stress on short-side:

[latex]\left(S_{m}\right)_{N}=\frac{P h}{2 t}=\frac{(150)(14)}{2(1)}=1050 psi[/latex]

11 . Total stress at corner of long-side:

[latex]\left(S_{t}\right)_{QM}=\left(S_{m}\right)_{M}+\left(S_{b}\right)_{Q M}=540+11,030=11,570 psi[/latex]

12 . Total stress at midpoint of long-side:

[latex]\left(S_{t}\right)_{M}=\left(S_{m}\right)_{M}+\left(S_{b}\right)_{M}=540+11,020=11,560 psi[/latex]

13 . Total stress at corner of short-side:

[latex]\left(S_{t}\right)_{QN}=\left(S_{m}\right)_{N}+\left(S_{b}\right)_{Q N}=1050+11,030=12,080 psi[/latex]

14 . Total stress at midpoint of short-side:

[latex]\left(S_{t}\right)_{N}=\left(S_{m}\right)_{N}+\left(S_{b}\right)_{N}=1050+5120=6170 psi[/latex]

All stresses are less than allowable stress values, so 1 -in. thickness is satisfactory.

Question 17.7: Determine the adequacy of a rectangular cross-sectional head...

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