Rivets
Introduction
Rivets are considered to
be permanent fasteners. Riveted joints are therefore similar to
welded and adhesive joints. When considering the strength of
riveted joints similar calculations are used as for bolted joints.
Rivets have been used in many large scale applications including shipbuilding,
boilers, pressure vessels, bridges and buildings etc. In recent
years there has been a progressive move from riveted joints to welded, bonded
and even bolted joints A riveted joint, in larger quantities is
sometimes cheaper than the other options but it requires higher skill levels
and more access to both sides of the joint
There are strict standards and codes for riveted joints used for
structural/pressure vessels engineering but the standards are less rigorous for
using riveted joints in general mechanical engineering.
A rivet is a cylindrical body called a shank with a head. A hot
rivet is inserted into a hole passing through two clamped plates to be attached
and the head is supported whilst a head is formed on the other end of the shank
using a hammer or a special shaped tool. The plates are thus permanently
attached. Cold rivets can be used for smaller sizes the  forming processes
being dependent on the ductility of the rivet material...
When a hot rivet cools it contracts imposing a compressive (clamping) stress on the plates. The rivet itself is then in tension the tensile stress is approximately equal to the yield stress of the rivet material
Strength of riveted joint
The notes below are assuming that the plate loads are withstood by the rivets. In practice the loads are generally withstood by friction between the plates under the compressive force of the contracted rivets. The calculations provided below are simplified but provide relatively conservative joint strength value. There is still a need to complete fatigue assessments on joints when relevant
Joint Types
There are two basic types of axial riveted joint the lap joint and the butt joint.
The selection of the number of rivets used for a joint and the array is simply to ensure the maximum strength of the rivets and the plates. If ten small arrayed rivets on a lap joint were replaced by three large rivets across a plate the plate section area (in tension) would clearly be significantly reduced...
Rivets for mechanical and structural applications are normally made from ductile (low carbon ) steel or wrought iron. For applications where weight, corrosion, or material constraints apply, rivets can be made from copper (+alloys) aluminum (+alloys),monel etc.
For rivets used for
structures and vessels etc the relevant design stresses are provided in the
applicable codes. For rivets used in mechanical engineering, values are
available in mechanical equipment standards which can be used with judgment.
BS 2573 Pt 1 Rules for the design of cranes includes design stress values based
on the Yields stress (0,2% proof stress) Y_{R0.2} as follows:
Hand driven rivets ..tensile stress (40%Y_{R0.2}) ..Shear (36,6%Y_{R0.2})..Bearing
(80%Y_{R0.2})
Machinery's handbook includes some values for steel rivets . I have interpreted
these values and include them below as rough approximate values for first
estimate. These are typical values for ductile steel.
Tensile (76MPa) .. Shear (61MPa) ..Bearing (131MPa)
In designing rivet joints it is convenient to simplify the process by making the following assumptions.

A rivet joint may fail as a result of one (or more) of a number mechanisms..

The rivet joint efficiency is simply described as follows
Eff = Max Allowable Force applied to Rivet Joint/ Plate Strength with no holes
The joint efficiency is increased by having multiple rows of rivets. It is also clear that the efficiency can never be 100%. The maximum allowable force is the smallest of the allowable shear, tensile or bearing forces.
Rivets are initially
sized with nominal diameters of between 1,2√ t and 1,4 √t (t =
plate thickness) The diametrical clearance provided for hot rivets
is about 1,5 mm max. For cold rivets very tight fits are often
provided by using reamed holes. It is important that the rivets are
not positioned too close to the side of the plate or the edge of a plate. m _{t}
should be greater than 1,5 d and m _{a} should be greater than 1,5 d.
(d = nominal rivet diameter). It is also suggested that the
distance between rivets in the rows (p_{t}) is greater than 3d and the
distance between rows (p_{a}) is greater than 3 d. Using
these guidance factors the strength calculations are simplified
Rivets calculations are generally completed to check for three failure modes:
rivet shear, plate tensile failure, and rivet /plate bearing. These are shown
below. A rivet joint can also fail due to plate shearing (tearing) behind
the rivet. This calculation is not always completed because the
joint design should include that the minimum metal land behind the rivet (m_{a}
above) is specified ensuring that other failure modes will operate before this
mode.
It is also important that the axial pitch (p _{a}) is maximized (see
above note) to ensure that the weakest section of the plate is through a row of
holes
Rivet Shear
The rivet shear calculation is
τ = F_{s} / ( n p d ^{2} /4 )
F_{s} = τ ( n p d ^{2} /4 )

Plate Tensile Stress
The tensile stress in the plate =
σ_{t} = F_{t} / [t (w  n _{r} d _{1})]
F_{t} = σ_{t} [t (w  n _{r} d _{1})]

Plate /rivet bearing stress
The Plate/Rivet bearing stress =
σ_{c} = F_{c} /(n d t)
F_{c} = σ_{c} (n d t)

Plate tearing stress..
The plate shear stress =
τ_{c} = F /(2 m _{a} t)
Determine the rivet joint efficiency:
Eff = Max Allowable Force applied to Rivet Joint/ Plate Strength with no holes
1 Shearing efficiency of the rivets:
Eff_{1} = τ (n p d ^{2} /4) / σ_{t }t w
Eff_{1} = τ (n p d ^{2} /4) / σ_{t }t w (1)
2 Tensile efficiency of the plate:
Eff_{2} = σ_{t} [t (w  n _{r} d _{1})] / σ_{t }t w
Eff_{2} = (w  n _{r} d _{1}) / w (2)
3 Crushing or bearing efficiency of the plate/rivets
Eff_{3} = σ_{c} (n d t) / σ_{t }t w
Eff_{3} = σ_{c} (n d) / σ_{t }w (3)
* Note: The minimum efficiency of the rivet joint is the minimum efficiency in the above cases.