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INPUT   DATA

♥ EXAMPLE Of Input/Output

Title

Nominal diameter of cap screw, d		in
X-section area of threaded portion of cap screw, AT		in2
Modulus of elasticity of cap screw, Eb		106lb/in2
Thickness of washer, tw		in
Diameter of washer, D		in
Thickness of member #1, t1		in
Modulus of elasticity of member #1, E1		106lb/in2
Thickness of member #2, t2		in
Modulus of elasticity of member #2, E2		106lb/in2
Cone angle of frustrum, α		°

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OUTPUT   VARIABLES   &   GRAPHS

STIFFNESS	Values	Units
♦ Threaded portion of cap screw, kT		lb/in
♦ Overall stiffness of cap screw, kcap screw		lb/in
♦ Stiffness of frustum #1, k1		lb/in
♦ Stiffness of middle frustum, km		lb/in
♦ Stiffness of frustum #2, k2		lb/in
♦ Overall stiffness of all frusta, kmembers		lb/in

♦ Middle frustum thickness, tmiddle

in

, from member #

Elastic Properties Of A Bolted Connection.

Consider a bolt joint clamping two members as shown. The stiffness of the cap screw and nut assembly can be approximated using the Frusta of A Hollow Cone model. The model assumes that the stresses are contained within the two conical frusta symmetrical about the the midplane of the joint each having a vertex angle of 2α. Depending mainly on the thicknesses of the members, the midplane splits one of the two members into two frusta, thus making a total of three frusta to analyse. Note that in the calculation, the thickness of member #2 is assumed to be h, the effective depth of screw threads into thickness of member #2.

The stiffness of each frustum is computed separately, and these are then aggregated via a series relation to obtain the overall joint stiffness. The stiffness of the bolt itself is computed by a similar series aggregation of the individual stiffnesses of the threaded and unthreaded portions. The relevant equations are given below:

where
     d = nominal diameter of the bolt
     D = diameter of the washer
     D^* = special diameter of the middle frustum
     h = effective thickness of member #2 (derived)
     L_g = grip length of the assembly (derived)
     t_n = thickness of member n
     t_w = thickness of the washer
     t_m = thickness of the middle frustum (derived)
     E_n = modulus of elasticity of component n
     α = cone angle of the frustum
     k_n = stiffness of component n

h is derived according to the following logic:
     if t₂ < d ⇒ h = t₂/2;
     if t₂ >= d ⇒ h = d/2;

Tips

    ◊ Use link EXAMPLE Of Input/Output  to demo data entry expectations and results; you may edit & use it as starting point

BIBLIOGRAPHY

• Brown TH; Marks' Calculations for Machine Design; McGraw Hill, New York, 2005, Chapter 8.
• Hicks TG; Mechanical Engineering Formulas, Pocket Guide; McGraw Hill, New York, 2003, Section 3.
• Young WC & Budynas RG; Roark's Formulas for Stress and Strain; 7th Edition, McGraw Hill, New York, 2002.