Problem 2. A total of 12 hexagonal bolt-nut assemblies, shown in the figure, are

Question

Problem 2. A total of 12 hexagonal bolt-nut assemblies, shown in the figure, are used to fasten a cylinder head onto a pressure vessel. The steel bolts are M12 ISO 9.8 with coarse thread, total length is 55 mm, threaded length is 30 mm, unthreaded length is 25 mm, major diameter is 12 mm, tensile-stress area is 84.3 mm2, minimum proof strength is 650 MPa, and elastic modulus is 207 GPa. The diameter of the bolt head and the nut is 18 mm. There are no washers in the assemblies and the grip length is 40 mm as shown in the figure. The pressure vessel is grade 30 cast iron with an elastic modulus of 100 GPa. The cvlinder head is steel with an elastic modulus of 207 GPa. 20 mm 20 mm 100 mm a) What is the stiffness of the bolt? b) What is the stiffness of the members? Use a-30° c) What is the stiffness constant of the joint? d) What is the external load per bolt due to a chamber pressure of 18 MPa? e) What is the recommended preload for a nonpermanent connection? D What is the safety factor against separation?

Explanation / Answer

Answer for (a)

The stiffness of the shank is given by:

Ks = AsEbolt/Ls

Ks = Stiffness of Shank

As= nominal area of bolt = Major diameter - 2 * depth of thread (Assume = 2*pitch)

For coarse pitch, assume depth of thread = 1.5 mm

As = ?(12 - 3)2/4 = 63.62 mm2

Ebolt = Elastic modulus of bolt

Ls = Length of shank = Unthreaded Length = 25mm

Here, Ebolt = 207 GPa Ls = 25mm As = 63.62 mm2

Therefore, Ks = 526.77 KN/mm

The stiffness of the threaded portion is given by

Kt = AtEbolt/Lt

At = tensile stress area = 84.3 mm2

Lt = Length of the thread in the grip = Lg - Ls = total grip (40mm) - length of shank (25mm) = 15mm

Therfore, Kt = 1163.34 KN/mm

The stiffness of bolt in the grip region is given by

Kbolt = KsKt/Ks+Kt = 362.58 KN/mm

Answer for (b)

Member stiffness is given by

Km = ?Edtan?/2ln{(L tan? + dw – d)(dw+d)/ (L tan? + dw + d)(dw-d)} ( Ref : Shigley and Mischke)

Where L = Grip length of the joint = 40mm

           d = bolt diameter = 12mm

           dw = diameter of the washer (since no washer is used, bolt head diameter is assumed as dw = 18mm)

           Ed = Difference in young's modulus of members = Esteel - Ecastiron = 207 - 100 = 107 GPa

           ? = 30 deg. Therefore tan? = 0.577

          Therefore, combined stiffness of the member, Km = 96.269 KN/mm

          

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