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**Stresses in screw fastenings**

It is necessary to determine the stresses in screw fastening due to both static and dynamic loading in order to determine their dimensions. In order to design for static loading both initial tightening and external loadings need be known.

**Initial tightening load **

When a nut is tightened over a screw following stresses are induced:

(a) Tensile stresses due to stretching of the bolt

(b) Torsional shear stress due to frictional resistance at the threads.

(c) Shear stress across threads

(d) Compressive or crushing stress on the threads

(e) Bending stress if the surfaces under the bolt head or nut are not perfectly normal to the bolt axis.

**(a) Tensile stress**

Since none of the above mentioned stresses can be accurately determined bolts are usually designed on the basis of direct tensile stress with a large factor of safety. The initial tension in the bolt may be estimated by an empirical relation P_{1}=284 d kN, where the nominal bolt diameter d is given in mm. The relation is used for making the joint leak proof. If leak proofing is not required half of the above estimated load may be used. However, since initial stress is inversely

proportional to square of the diameter bolts of smaller diameter such as M16 or M8 may fail during initial tightening. In such cases torque wrenches must be used to apply known load.

The torque in wrenches is given by T= C P_{1}d where, C is a constant depending on coefficient of friction at the mating surfaces, P_{1} is tightening up load and d is the bolt diameter.

(b) Torsional shear stress This is given by where T is the torque and d the core diameter. We may relate torque T to the tightening load P1 in a power screw configuration **(figure-4.4.1.1.1 )** and taking collar friction into account we may write

where d_{m } and d_{cm} are the mean thread diameter and mean collar diameter respectively, and m μ and μ_{c} are the coefficients of thread and collar friction respectively α is the semi thread angle. If we consider that

then we may write T= C P_{1}d m where C is a constant for a given arrangement. As discussed earlier similar equations are used to find the torque in a wrench.

**(c) Shear stress across** the threads This is given by where d is the core diameter and b is the base width of the thread and n is the number of threads sharing the load

**(d) Crushing stress on threads** This is given by where and d are the outside and core diameters as shown in **figure- 4.4.1.1.1**

**(e) Bending stress **If the underside of the bolt and the bolted part are not parallel as shown in figure4.4.1.1.2, the bolt may be subjected to bending and the bending stress may be given by where x is the difference in height between the extreme corners of the nut or bolt head, L is length of the bolt head shank and E is the young’s modulus.

**4.4.1.1.2F- Development of bending stress in a bolt**

**Stresses due to an external load **

If we consider an eye hook bolt as shown in **figure- 4.4.1.2.1** where the complete machinery weight is supported by threaded portion of the bolt, then the bolt is subjected to an axial load and the weakest section will be at the root of the thread. On this basis we may write

where for fine threads d_{c} =0.88d and for coarse threads dc =0.84d, d being the nominal diameter.

Bolts are occasionally subjected to shear loads also, for example bolts in a flange coupling as shown in **figure- 4.4.1.2.2.** It should be remembered in design that shear stress on the bolts must be avoided as much as possible. However if this cannot be avoided the shear plane should be on the shank of the bolt and not the threaded portion. Bolt diameter in such cases may be found from the relation

where n is the number of bolts sharing the load, τ is the shear yield stress of the bolt material. If the bolt is subjected to both tensile and shear loads, the shank should be designed for shear and the threaded portion for tension. A diameter slightly larger than that required for both the cases should be used and it should be checked for failure using a suitable failure theory.

**4.4.1.2.2F- A typical rigid flange coupling**

**Combined effect of initial tightening load and external load **

When a bolt is subjected to both initial tightening and external loads i.e. when a preloaded bolt is in tension or compression the resultant load on the bolt will depend on the relative elastic yielding of the bolt and the connected members. This situation may occur in steam engine cylinder cover joint for example. In this case the bolts are initially tightened and then the steam pressure applies a tensile load on the bolts. This is shown in figure-4.4.1.3.1 (a) and 4.4.1.3.1 (b).

**4.4.1.3.1F- A bolted joint subjected to both initial tightening and external load**

Initially due to preloading the bolt is elongated and the connected members are compressed. When the external load P is applied, the bolt deformation increases and the compression of the connected members decreases. Here P_{1} and P_{2} in figure 4.4.1.3.1 (a) are the tensile loads on the bolt due to initial tightening and external load respectively. The increase in bolt deformation is given by ** **and decrease in member compression is where, P_{b} is the share of P_{2} in bolt, P_{C} is the share of P_{2} in members, K_{b} and K_{c} are the stiffnesses of bolt and members. If the parts are not separated then and this gives

Therefore, the total applied load P_{2} due to steam pressure is given by

P_{2}= P_{b}+P_{c }

This gives P_{b}= P _{2 }K, where . Therefore the resultant load on bolt is P_{1} +KP_{2} . Sometimes connected members may be more yielding than the bolt and this may occurs when a soft gasket is placed between the surfaces. Under these circumstances and this gives K≈ 1. Therefore the total load P = P_{1} + P_{2}

Normally K has a value around 0.25 or 0.5 for a hard copper gasket with long through bolts. On the other hand if K_{c} >>K_{b} , K approaches zero and the total load P equals the initial tightening load. This may occur when there is no soft gasket and metal to metal contact occurs. This is not desirable. Some typical values of the constant K are given in table 4.4.1.3.1.

Type of joint | K |

Metal to metal contact with through bolt Hard copper gasket with long through bolt Soft copper gasket with through bolts Soft packing with through bolts Soft packing with studs | 0-0.1 0.25-0.5 0.50- 0.75 0.75- 1.00 1.00 |

**Leak proof joint**

The above analysis is true as long as some initial compression exists. If the external load is large enough the compression will be completely removed and the whole external load will be carried by the bolt and the members may bodily separate leading to leakage. Therefore, the condition for leak proof joint is Substituting P_{b}=P_{2} K and

the condition for a leak proof joint reduces to P_{1} >P_{2} (1-K). It is therefore necessary to maintain a minimum level of initial tightening to avoid leakage.

**Joint separation**

Clearly if the resultant load on a bolt vanishes a joint would separate and the condition for joint separating may be written as P_{1}+KP_{2} =0

Therefore if P_{1} >KP _{2} and P_{1}< A_{b} σ_{tyb} , there will be no joint separation. Here A_{b} and σ_{tyb }are the bolt contact area and tensile yield stress of the bolt material respectively and condition ensures that there would be no yielding of the bolt due to initial tightening load.

The requirement for higher initial tension and higher gasket factor (K) for a better joint may be explained by the simple diagram as in figure- 4.4.3.1.

**4.4.3.1F – Force diagram for joint separation**

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