Bearing pressure is a particular case of
contact mechanics
Contact mechanics is the study of the deformation of solids that touch each other at one or more points.Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.Popov, Valentin L., 2010, ''Contact Mechanics and Friction. Physical Pri ...
often occurring in cases where a convex surface (male cylinder or sphere) contacts a concave surface (female cylinder or sphere:
bore
Bore or Bores often refer to:
*Boredom
* Drill
Relating to holes
* Boring (manufacturing), a machining process that enlarges a hole
** Bore (engine), the diameter of a cylinder in a piston engine or a steam locomotive
** Bore (wind instruments), ...
or
hemispherical cup). Excessive contact pressure can lead to a typical bearing failure such as a plastic deformation similar to
peening
In metallurgy, peening is the process of working a metal's surface to improve its material properties, usually by mechanical means, such as hammer blows, by blasting with shot (shot peening), or focusing light (laser peening). Peening is norma ...
. This problem is also referred to as bearing resistance.
[EN 1993-1-8:2005 '' Eurocode 3: Design of steel structures - Part 1-8: Design of joints'']
Hypotheses
A contact between a male part (convex) and a female part (concave) is considered when the radii of curvature are close to one another. There is no tightening and the joint slides with no friction therefore, the contact forces are
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
to the tangent of the contact surface.
Moreover, bearing pressure is restricted to the case where the charge can be described by a radial
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
pointing towards the center of the joint.
Case of a cylinder-cylinder contact
In the case of a
revolute joint
A revolute joint (also called pin joint or hinge joint) is a one- degree-of-freedom kinematic pair used frequently in mechanisms and machines. The joint constrains the motion of two bodies to pure rotation along a common axis. The joint does no ...
or of a
hinge joint
A hinge joint (ginglymus or ginglymoid) is a bone joint in which the articular surfaces are molded to each other in such a manner as to permit motion only in one plane. According to one classification system they are said to be uniaxial (having ...
, there is a contact between a male cylinder and a female cylinder. The complexity depends on the situation, and three cases are distinguished:
* the
clearance is negligible:
** a) the parts are
rigid bodies
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
,
** b) the parts are
elastic bodies;
* c) the clearance cannot be ignored and the parts are elastic bodies.
By "negligible clearance", H7/g6
fit is typically meant.
The axes of the cylinders are along the ''z''-axis, and two external forces apply to the male cylinder:
* a force
along the ''y''-axis, the load;
* the action of the bore (contact pressure).
The main concern is the contact pressure with the bore, which is uniformly distributed along the ''z''-axis.
Notation:
* ''D'' is the nominal diameter of both male and female cylinders;
* ''L'' the guiding length.
Negligible clearance and rigid bodies
In this first modeling, the pressure is uniform. It is equal to
:
:
.
Negligible clearance and elastic bodies
If it is considered that the parts deform elastically, then the contact pressure is no longer uniform and transforms to a sinusoidal repartition
:
: ''P''(θ) = ''P''
max⋅cos θ
with
:
.
This is a particular case of the following section (θ
0 = π/2).
The maximum pressure is 4/π ≃ 1.27 times bigger than the case of uniform pressure.
Clearance and elastic bodies
In cases where the clearance can not be neglected, the contact between the male part is no longer the whole half-cylinder surface but is limited to a 2θ
0 angle. The pressure follows
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
:
: ''P''(θ) = ''K''⋅δ
α(θ)
where
* ''K'' is a positive real number that represents the rigidity of the materials;
* δ(θ) is the radial displacement of the contact point at the angle θ;
* α is a coefficient that represents the behaviour of the material:
** α = 1 for metals (purely
elastic behaviour),
** α > 1 for polymers (
viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
or
viscoplastic
Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The in ...
behaviour).
The pressure varies as:
: ''A''⋅cos θ - ''B''
where ''A'' and ''B'' are positive real number. The maximum pressure is:
:
the angle θ
0 is in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s.
The rigidity coefficient ''K'' and the half contact angle θ
0 can not be derived from the theory. They must be measured. For a given system — given diameters and materials —, thus for given ''K'' and clearance ''j'' values, it is possible to obtain a curve θ
0 = ƒ(''F''/(''DL'')).
Case of a sphere-sphere contact
A sphere-sphere contact corresponds to a
spherical joint
In an automobile, ball joints are spherical bearings that connect the control arms to the steering knuckles, and are used on virtually every automobile made. They bionically resemble the ball-and-socket joints found in most tetrapod animals. ...
(socket/ball), such as a ball jointed cylinder saddle. It can also describe the situation of
bearing balls.
Case of uniform pressure
The case is similar as above: when the parts are considered as rigid bodies and the clearance can be neglected, then the pressure is supposed to be uniform. It can also be calculated considering the projected area
:
:
.
Case of a sinusoidal repartition of pressure
As in the case of cylinder-cylinder contact, when the parts are modeled as elastic bodies with a negligible clearance, then the pressure can be modeled with a sinusoidal repartition
:
: ''P''(θ, φ) = ''P''
max⋅cos θ
with
:
.
Hertz contact stress
When the clearance can not be neglected, it is then necessary to know the value of the half contact angle θ
0 , which can not be determined in a simple way and must be measured. When this value is not available, the Hertz contact theory can be used.
The Hertz theory is normally only valid when the surfaces can not conform, or in other terms, can not fit each other by elastic deformation; one surface must be convex, the other one must be also convex plane. This is not the case here, as the outer cylinder is concave, so the results must be considered with great care. The approximation is only valid when the inner radius of the container ''R''
1 is far greater than the outer radius of the content ''R''
2, in which case the surface container is then seen as flat by the content. However, in all cases, the pressure that is calculated with the Hertz theory is greater than the actual pressure (because the contact surface of the model is smaller than the real contact surface), which affords designers with a safety margin for their design.
In this theory, the radius of the female part (concave) is negative.
A relative diameter of curvature is defined:
:
where ''d''
1 is the diameter of the female part (negative) and ''d''
2 is the diameter of the male part (positive). An equivalent module of elasticity is also defined:
:
where ν
''i'' is the
Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
of the material of the part ''i'' and ''E
i'' its
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
.
For a cylinder-cylinder contact, the width of the contact surface is:
:
and the maximal pressure is in the middle:
:
.
In case of a sphere-sphere contact, the contact surface is a disk whose radius is:
:
and the maximal pressure is in the middle:
: