25-Nav-B1 Applied Thermodynamics and Heat Transfer (25-Mec-A1) · December 2013
Question 3 of 6: Bolted-Joint Design and Optimum Preload
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2013 · 98-Mar-B1 Advanced Machine Design.
Open-book, 3 hours, 100 marks. The paper requires all of Part I (Problems 1 and 2)
plus any three of the four Part II problems (3–6). All six problems are solved in full below.
Reference texts. R. G. Budynas & J. K. Nisbett, Shigley’s
Mechanical Engineering Design, 10th ed. (shafts & critical speed §7; bolted joints
§8; lubrication & journal bearings §12; brakes §16); R. C. Juvinall &
K. M. Marshek, Fundamentals of Machine Component Design (bearings, brakes, impact);
R. C. Hibbeler, Mechanics of Materials (bending, impact loading).
Check: this paper, although listed under Applied Thermodynamics and Heat Transfer, is headed “98-Mar-B1” with printed title “Advanced Machine Design”; it is a general mechanical machine-design paper — stress/yield criteria, shaft deflection/critical speed, bolted-joint preload, journal-bearing sizing, impact loading, and drum brakes — with zero thermodynamics or heat-transfer content, and it is solved as the exam actually printed.
Question 3: Bolted-Joint Design and Optimum Preload (20 marks)
Given. A single-bolt tension joint clamping two steel members, grip length
$l = 2$ in, member plan dimension $D = 1$ in, external separating load $P = 2000$ lb.
Bolted joint: bolt of diameter $d$ through two steel members of
combined grip $l = l_1+l_2 = 2$ in; external load $P$ pulls the members apart.
Find. A suitable bolt size and preload, the factors of safety against bolt yielding and
against joint separation, and the optimum preload (as a fraction of proof strength) that maximizes those
factors.
Approach. Compute the bolt and member stiffnesses to get the joint stiffness constant
$C$, express the yielding and separation factors as functions of preload, then choose the preload that makes
them equal (the optimum) and select a standard bolt/grade that clears the required load.
Select a trial bolt. Take a $\tfrac12$–13 UNC SAE Grade 5 bolt: tensile-stress
area $A_t = 0.1419\ \text{in}^2$, proof strength $S_p = 85\ \text{ksi}$, so the proof load is
$F_p = S_pA_t = 12{,}060$ lb. Its major-diameter area is $A_d = \pi d^2/4 = 0.1963\ \text{in}^2$.
Bolt stiffness. For a 2.5-in bolt the threaded length in the grip is
$l_t = l-l_d = 0.75$ in and the shank portion $l_d = 1.25$ in, so
$$k_b = \dfrac{A_dA_tE}{A_dl_t+A_tl_d} = 2.58\times10^{6}\ \text{lb/in}.$$
Member stiffness. Using the Wileman relation for a steel joint,
$k_m = E\,d\,(0.78715)\exp(0.62873\,d/l) = 1.38\times10^{7}\ \text{lb/in}$. The members are much stiffer than
the bolt, which is the desirable condition.
Joint constant.
$$C = \dfrac{k_b}{k_b+k_m} = \dfrac{2.58}{2.58+13.8} = \boxed{0.157}.$$
Only this fraction of the external load $P$ reaches the bolt; the rest unloads the members.
Optimum preload. The yielding factor is
$n_p = (S_pA_t-F_i)/(CP)$ and the separation factor is $n_0 = F_i/[P(1-C)]$. The first falls and the
second rises with preload $F_i$, so the pair is maximized where they are equal. Setting $n_p=n_0$ and
solving,
$$F_i = (1-C)\,S_pA_t \;\Rightarrow\; \dfrac{F_i}{S_pA_t} = 1-C = \boxed{0.843}.$$
The optimum preload is therefore 84% of proof strength, i.e.
$F_i = 0.843(12{,}060) \approx 10{,}170$ lb — within the range permitted for a permanent
connection (up to 90% of proof).
Resulting factors of safety. At the optimum both factors collapse to the same clean
expression,
$$n_p = n_0 = \dfrac{S_pA_t}{P} = \dfrac{12{,}060}{2000} = \boxed{6.0}.$$
The generous margin confirms the $\tfrac12$-in Grade 5 bolt is more than adequate; a smaller bolt could
be used if a lower factor were acceptable.
