A 1.75 mm pitch x 12 mm diameter grade 8.8 bolt is used to fix a part. Assume
that the steel part has a cross section area of 200 mm^{2} and is subjected
to a cyclic load varying between 0 and 20000 N. It is assumed that the bolt threads
give rise to a stress concentration factor of 3.

**Determine:** a) The FoS of bolt when there is no pre-load.

b) The minimum required pre-load, F_{o} to prevent loss of compression.

c) The FoS for the bolt when the pre-load, F_{o} = 22000 N.

d) The minimum force in the part when the pre-load is 22000 N.

**Solution:**

It should be noted that from BS 6104, Pt 1, 1981, Table 6, the ultimate tensile load of
this size and type bolt is given as 67,400 N. The bolt core area is 84.3 mm^{2}

The UTS is 800 MPa, we assume that the yield strength is 500 MPa and that the fatigue
endurance strength is half the UTS, ie 400 MPa.

**a)** FoS with zero pre-load.

To determine the FoS the mean stress and the stress amplitude, multiplied by the
stress concentration factor, will be plotted on a modified Goodman diagram. With no
pre-load the entire load is carried by the bolt.
For a load varying between zero and 20000 N, the mean is 10000 N and
the amplitude is 10000 N.

The mean stress, S_{m} = 20000 /(2 x 84.3) = 118.6 MPa.

The amplitude of the stress multiplied by the stress concentration factor (3) is 355.9 MPa.

When this is plotted on the modified Goodman diagram, the point A falls above and to the
right of the line, outside the safe zone, the design is unsafe, the FoS, based on
the yield strength, is given by the ratio of the lengths of the lines OB / OA or
algebraically from the modified Goodman diagram:

1/FoS = (118.6 / 500) + (3 x 118.6 / 400)

FoS = 0.89

**b)** The minimum F_{o} required to prevent loss of compression.

For bodies of equal length and equal modulii, the spring constants are proportional to the csa.
The csa of the bolt is 84.3 mm^{2}.

When the part has zero compression:

**c)** Find the FoS in the bolt when F_{o} = 22000 N

F_{b mean} = (10000 x 84.3 / 284.3) + 22000 = 24970 N

stress_{mean} = 24970 / 84.3 = 296.1 MPa

F_{b amplitude} = k_{b} P_{a}/(k_{p} + k_{b})
=84.3 x 10000 / 284.3 = 2965 N

stress_{amplitude} = 2965 / 84.3 = 35.17 MPa

The effect of the pre-load is to greatly reduce the magnitude of the
alternating force and stress in the bolt (which is normally much more critical) while increasing the
mean bolt force and stress (normally less critical) resulting in the bottom diagram.

**d)** The minimum pre-load in the part when the bolt pre-load,
F_{o} = 22000 N.

F_{p min} = (200 x 20000 / 284.3) - 22000 = -7930 N

Return to Module Home Page

David J Grieve, 10th February 2004.