HOME

TheInfoList



OR:

Bearing pressure is a particular case of
contact mechanics Contact mechanics is the study of the Deformation (mechanics), deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between Stress (mechanics), stresses acting perpendicular to the cont ...
often occurring in cases where a convex surface (male cylinder or sphere) contacts a concave surface (female cylinder or sphere: bore 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), focusing light ( laser peening), or in recent years ...
. 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 cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
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-Degrees of freedom (mechanics), degree-of-freedom kinematic pair used frequently in Mechanism (engineering), mechanisms and machines. The joint Constraint (classical mechanics), co ...
or of a
hinge joint A hinge joint (ginglymus or ginglymoid) is a bone joint where 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 o ...
, 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 negligible, when a deforming pressure or deforming force is applied on it. The distance between any two given points on a rigid body rema ...
, ** 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 \vec 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: : P = \frac = \frac.


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 : P_\mathrm = \frac \cdot \frac. This is a particular case of the following section (θ0 = 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 2θ0 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 by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
: : ''P''(θ) = ''K''⋅δα(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 Viscosity, viscous and Elasticity (physics), elastic characteristics when undergoing deformation (engineering), deformation. Viscous mate ...
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 i ...
behaviour). The pressure varies as: : ''A''⋅cos θ - ''B'' where ''A'' and ''B'' are positive real number. The maximum pressure is: : P_\mathrm = \frac \times \frac the angle θ0 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. It is defined such that one radian is the angle subtended at ...
s. The rigidity coefficient ''K'' and the half
contact angle The contact angle (symbol ) is the angle between a liquid surface and a solid surface where they meet. More specifically, it is the angle between the surface tangent on the liquid–vapor interface and the tangent on the solid–liquid interfac ...
θ0 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 θ0 = ƒ(''F''/(''DL'')).


Case of a sphere-sphere contact

A sphere-sphere contact corresponds to a
spherical joint A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center'' ...
(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 Projected area is the two dimensional area measurement of a three-dimensional object by projecting its shape on to an arbitrary plane. This is often used in mechanical engineering and architectural engineering related fields, especially for hard ...
: : P = \frac = \frac.


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 : P = \frac \cdot \frac.


Hertz contact stress

When the clearance can not be neglected, it is then necessary to know the value of the half contact angle θ0 , 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: : \frac = \frac + \frac 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: : \frac = \frac + \frac where ν''i'' is the
Poisson's ratio In materials science and solid mechanics, Poisson's ratio (symbol: ( 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 the material of the part ''i'' and ''Ei'' its
Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
. For a cylinder-cylinder contact, the width of the contact surface is: : b = \left ( \frac \cdot \frac \right )^ and the maximal pressure is in the middle: : P_\max = \frac = \sqrt. In case of a sphere-sphere contact, the contact surface is a disk whose radius is: : a = \left ( \frac \cdot \frac \right )^ and the maximal pressure is in the middle: : P_\max = \frac = \frac \sqrt /math>.


Applications


Bolt used as a stop

In a bolted connection, the role of the bolts is normally to press one parts on the other; the adherence (
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
) is opposed to the tangent forces and prevents the parts from sliding apart. In some cases however, the adherence is not sufficient. The bolts then play the role of stops: the screws endure
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
whereas the hole endure bearing pressure. In order to increase the bearing pressure of a material, there are several factors that can be considered. One of the most effective methods is to increase the surface area of the material. By increasing the surface area, the load is distributed over a larger area, reducing the bearing pressure. In good design practice, the threaded part of the screw should be small and only the smooth part should be in contact with the plates; in the case of a shoulder screw, the clearance between the screw and the hole is very small ( a case of rigid bodies with negligible clearance). If the acceptable pressure limit ''P''lim of the material is known, the thickness ''t'' of the part and the diameter ''d'' of the screw, then the maximum acceptable tangent force for one bolt ''F''b, Rd (design bearing resistance per bolt) is: : ''F''b, Rd = ''P''lim × ''d'' × ''t''. In this case, the acceptable pressure limit is calculated from the ultimate tensile stress ''f''u and factors of safety, according to the Eurocode 3 standard. In the case of two plates with a single overlap and one row of bolts, the formula is: : ''P''lim = 1.5 × ''f''uM2 where * γM2 = 1.25: partial safety factor. In more complex situations, the formula is: : ''P''lim = ''k''1 × α × ''f''uM2 where * ''k''1 and α are factors that take into account other failure modes than the bearing pressure overload; ''k''1 take into account the effects that are perpendicular to the tangent force, and α the effects along the force; * ''k''1 = min for end bolts,
''k''1 = min for inner bolts, ** ''e''2: edge distance from the centre of a fastener hole to the adjacent edge of the part, measured at right angles to the direction of load transfer, ** ''p''2: spacing measured perpendicular to the load transfer direction between adjacent lines of fasteners, ** ''d''0: diameter of the passthrough hole; * α = min, with ** ''e''1: end distance from the center of a fastener hole to the adjacent end of the part, measured in the direction of load transfer, ** ''p''1: spacing between centers of fasteners in the direction of load transfer, ** ''f''ub: specified
ultimate tensile strength Ultimate tensile strength (also called UTS, tensile strength, TS, ultimate strength or F_\text in notation) is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials, the ultimate t ...
of the bolt. When the parts are in wood, the acceptable limit pressure is about 4 to 8.5 MPa.


Plain bearing

In
plain bearing file:NYC 100-driving-axle-friction-bearing.jpg, Plain bearing on a 1906 S-Motor locomotive showing the axle, bearing, oil supply and oiling pad file:Linear-table with detail numbered.png, A sliding table with four cylindrical bearings file:GWR Spo ...
s, the
shaft Shaft may refer to: Rotating machine elements * Shaft (mechanical engineering), a rotating machine element used to transmit power * Line shaft, a power transmission system * Drive shaft, a shaft for transferring torque * Axle, a shaft around whi ...
is usually in contact with a bushing (sleeve or flanged) to reduce
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
. When the rotation is slow and the load is radial, the model of uniform pressure can be used (small deformations and clearance). The product of the bearing pressure times the circumferential sliding speed, called load factor PV, is an estimation of the resistance capacity of the material against the frictional heating.


References


Bibliography

* ublin 1992 * hevalier 2004 * anchon 2001 * anchon 2011 * CM 2000 * G 2003 {{Authority control Bearings (mechanical) Mechanical engineering Solid mechanics