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22-Mec-B1 Advanced Machine Design · May 2018

Question 5 of 6: Bolted Pressure-Vessel Cylinder Head

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Exams, 16-Mec-B1 Advanced Machine Design, May 2018. Open book, 3 hours, 100 marks. Part I (Problems 1 & 2) is compulsory; candidates answer only three of the four Part II problems (3–6). All six problems are solved here as a complete study resource.

Reference texts. Budynas & Nisbett, Shigley’s Mechanical Engineering Design (10th ed.) — shaft/fatigue §7, bolted joints §8, journal bearings §12, brakes §16, power screws §8–2; Juvinall & Marshek, Fundamentals of Machine Component Design; Norton, Machine Design: An Integrated Approach.

Question 5: Bolted Pressure-Vessel Cylinder Head (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Ten M12 steel bolts clamp a steel head over a 150 mm sealing diameter; the members grip is $D+E=45\ \text{mm}$ of steel.

Given data
Sealing diameter / pressure$D_{\text{seal}}=150\ \text{mm}$, $p=6\ \text{MPa}$
Number / size of bolts10 × M12 ($A_t=84.3\ \text{mm}^2$)
Grip (steel head + flange)$l=D+E=20+25=45\ \text{mm}$
Required factors$n_{\text{sep}}\ge1.5$, $n_{\text{yield}}\ge2$
Steel modulus$E_{\text{steel}}=207\ \text{GPa}$

Find. The required bolt preload $F_i$ and a suitable metric bolt grade.

confined gasket (seal Ø 150 mm)M12 bolt (x10)internal gas pressure p = 6 MPaD = 20, E = 25 mm → grip 45 mm; A=100 B=200 C=300 mm
Bolted cylinder-head joint: 10 M12 bolts preloaded to seal against 6 MPa gas acting over the 150 mm sealing diameter.

Approach. Find the external load per bolt from the pressure on the sealing area, compute the joint stiffness constant $C$ from the bolt and member (frustum) stiffnesses, set the preload from the separation factor, then check which bolt grade satisfies the yield factor.

  1. External load per bolt. The gas force on the sealed area is shared by 10 bolts: $$P=\frac{p\,(\pi/4)D_{\text{seal}}^2}{n}=\frac{6\times10^{6}(\pi/4)(0.150)^2}{10}=\frac{106\,029}{10}=1.06\times10^{4}\ \text{N/bolt}.$$
  2. Joint stiffness constant. Bolt stiffness $k_b$ (shank + threaded lengths within the 45 mm grip) and member stiffness $k_m$ (Shigley steel frustum, $k_m=E d\,(0.78715)e^{0.62873\,d/l}$) give $$C=\frac{k_b}{k_b+k_m}=0.155.$$ Only a fraction $C=15.5\%$ of each external load reaches the bolt; the members carry the rest.
  3. Preload from separation. Separation occurs when the members unload, at $P_0=F_i/(1-C)$. Requiring $n_{\text{sep}}=1.5$: $$F_i=n_{\text{sep}}\,P\,(1-C)=1.5(10\,603)(1-0.155)=\boxed{13.4\ \text{kN}}.$$ This is 42% of the M12 proof load — comfortably within the usual $\le90\%$ preload band.
  4. Select bolt grade (yield check). The most-loaded bolt tension is $F_i+CP$; requiring $n_{\text{yield}}=2$ against proof needs $$S_p\ge\frac{F_i+n_{\text{yield}}CP}{A_t}=\frac{13\,441+2(0.155)(10\,603)}{84.3\times10^{-6}}=198\ \text{MPa}.$$ A metric Class 5.8 bolt ($S_p=380\ \text{MPa}$) easily satisfies this; its actual yield factor is $$n_{\text{yield}}=\frac{S_pA_t-F_i}{CP}=\frac{380(84.3)-13\,441}{0.155(10\,603)}=11.3\gg2\ \checkmark.$$
Problem 5 results
QuantityValue
Total gas force / per bolt$106.0\ \text{kN}$ / $P=10.6\ \text{kN}$
Joint stiffness constant$C = 0.155$
Required preload$F_i = 13.4\ \text{kN}$ (42% proof)
Selected gradeClass 5.8 ($S_p=380$ MPa); $n_{\text{yield}}=11.3$, $n_{\text{sep}}=1.5$
Check: the member stiffness uses the Shigley single-frustum steel model over the 45 mm grip; a full two-frustum integration changes $C$ by only a percent or two and does not alter the grade choice. Grade 5.8 is chosen as the lowest common metric grade meeting the modest $198\ \text{MPa}$ proof requirement with ample margin.