Question 10.22: Two steel members are joined by two M 12 × 1.5, class 5.8 bo......

For M12 × 1.5, class 5.8 bolts

A_t=88.1 \mathrm{~mm}^2, \sigma_p=380 \mathrm{~MPa}, \sigma_u=520 \mathrm{~MPa}

assuming rolled threads, k_f=2.2

Given \frac{k_b}{k_c}=\frac{1}{3}

where k_c is the combined stiffness and k_b is the bolt stiffness.

C=\frac{k_b}{k_b+k_c}=\frac{k_b}{k_b+3 k_b}=\frac{1}{4}=0.25

(a) We have F_b=F_i+\frac{k_b}{\left(k_c+k_b\right)} F

Load to cause joint separation

F_b=F

and the above equation becomes

F=F_i+\frac{k_b}{\left(k_c+k_b\right)} F

or F_i=F\left\{1-\frac{k_b}{\left(k_c+k_b\right)}\right\}

Hence to prevent separation

F_i>F\left\{1-\frac{k_b}{\left(k_c+k_b\right)}\right\}

F_i>12(1-0.25)=9 \mathrm{~kN}

(b) F_b=F_i+\frac{k_b}{\left(k_c+k_b\right)} F

\left(F_b\right)_{\max }=9+\frac{1}{4} \times 12=12 \mathrm{~kN}

\left(F_b\right)_{\min }=9+\frac{1}{4} \times 0=9 \mathrm{~kN}

\left(F_b\right)_m=\frac{9+12}{2}=10.5 \mathrm{~kN}

\left(F_b\right)_a=\frac{12-9}{2}=1.5 \mathrm{~kN}

\sigma_m=\frac{\left(F_b\right)_m}{A_t}=\frac{10.5 \times 1000}{88.1}=119.18 \mathrm{~MPa}

\sigma_a=\frac{\left(F_b\right)_a}{A_t}=\frac{1.5 \times 1000}{88.1} 17.02 \mathrm{~MPa}

S_e=\frac{176}{2.2}=80 \mathrm{~MPa}

\sigma_i=\frac{F_i}{A_t}=\frac{9000}{88.1}=102.16 \mathrm{~MPa}

S_m=\frac{\sigma_u\left(S_e+\sigma_i\right)}{S_e+\sigma_u}=\frac{520(80+102.16)}{80+520}=157.87 \mathrm{~MPa}

S_a=S_m-\sigma_i

S_a=157.87-102.16=55.712 \mathrm{~MPa}

Factor of safety against fatigue failure is (C = 1)

N=\frac{S_a}{\sigma_a}=\frac{55.712}{17.02}=3.273

Alternatively,

N=\frac{\sigma_u A_t-F_i}{\frac{C F}{2}\left(1+\frac{\sigma_u}{S_e}\right)}=\frac{520 \times 88.1-9000}{\frac{0.25 \times 12000}{2}\left(1+\frac{520}{80}\right)}=3.272

Factor of safety in the absence of preload (C = 1)

N_0=\frac{\sigma_u A_t}{\frac{F}{2}\left(1+\frac{\sigma_u}{S_e}\right)}=\frac{520 \times 88.1}{\frac{12000}{2}\left(1+\frac{520}{80}\right)}=1.018

It is clear that the presence of pre load is beneficial as it increases the factor of safety.