Bearing pressure

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 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 pointing towards the center of the joint.

Bearing pressure for a cylinder-cylinder contact.

In the case of a revolute joint or of a hinge joint, 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,
  • 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 ${\displaystyle {\vec {F}}}$ 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;[2]
• L the guiding length.

Negligible clearance and rigid bodies

Uniform bearing pressure: case of rigid bodies when the clearing can be neglected.

In this first modeling, the pressure is uniform. It is equal to[3] · [4] · :[5]

${\displaystyle P={\frac {F}{D\times L}}={\frac {\mathrm {radial\ load} }{\mathrm {projected\ area} }}}$.

Negligible clearance and elastic bodies

Bearing pressure with a sinusoid repartition: case of elastic bodies when the clearing can be neglected.

If it is considered that the parts deform elastically, then the contact pressure is no longer uniform and transforms to a sinusoidal repartition[6] · :[7]

P(θ) = P_max⋅cos θ

with

${\displaystyle P_{\mathrm {max} }={\frac {4}{\pi }}\cdot {\frac {F}{LD}}}$.

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

Bearing pressure in case of elastic bodies when the clearance must be taken into account.

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:[8]

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 or viscoplastic behaviour).

The pressure varies as:

A⋅cos θ - B

where A and B are positive real number. The maximum pressure is:

${\displaystyle P_{\mathrm {max} }={\frac {4F}{LD}}\times {\frac {1-\cos \theta _{0}}{2\theta _{0}-\sin 2\theta _{0}}}}$

the angle θ₀ is in radians.

The rigidity coefficient K and the half contact angle 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)).

Bearing pressure in the case of a sphere-sphere contact.

A sphere-sphere contact corresponds to a spherical joint (socket/ball), such as a ball jointed cylinder saddle. It can also describe the situation of bearing balls.

Hertz contact stress in the case of a male cylinder-female cylinder contact.

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 R₁ is far greater than the outer radius of the content R₂, 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.[12]

A relative diameter of curvature is defined:

${\displaystyle {\frac {1}{d^{*}}}={\frac {1}{d_{1}}}+{\frac {1}{d_{2}}}}$

where d₁ is the diameter of the female part (negative) and d₂ is the diameter of the male part (positive). An equivalent module of elasticity is also defined:

${\displaystyle {\frac {1}{E^{*}}}={\frac {1-\nu _{1}^{2}}{E_{1}}}+{\frac {1-\nu _{2}^{2}}{E_{2}}}}$

where ν_i is the Poisson's ratio of the material of the part i and E_i its Young's modulus.

For a cylinder-cylinder contact, the width of the contact surface is:

${\displaystyle b=\left({\frac {2F}{\pi L}}\cdot {\frac {d^{*}}{E^{*}}}\right)^{1/2}}$

and the maximal pressure is in the middle:

${\displaystyle P_{\max }={\frac {2F}{\pi bL}}={\sqrt {\frac {2FE^{*}}{\pi Ld^{*}}}}}$.

Hertz contact stress in the case of a male sphere-female sphere contact.

In case of a sphere-sphere contact, the contact surface is a disk whose radius is:

${\displaystyle a=\left({\frac {3F}{8}}\cdot {\frac {d^{*}}{E^{*}}}\right)^{1/3}}$

and the maximal pressure is in the middle:

${\displaystyle P_{\max }={\frac {3F}{\pi a^{2}}}={\frac {4}{\pi }}{\sqrt[{3}]{3F\left({\frac {E^{*}}{d^{*}}}\right)^{2}}}}$.

• [Aublin 1992] Aublin, Michel; Boncompain, René; Boulaton, Michel; Caron, Daniel; Jeay, Émile; Lacage, Bernard; Réa, Jacky (1992). Systèmes mécaniques : théorie et dimensionnement (in French). Dunod. pp. 108–157. ISBN 2-10-001051-4.
• [Chevalier 2004] Chevalier, André (2004). Guide du dessinateur industriel (in French). Hachette technique. p. 258. ISBN 978-2-01-168831-6.CS1 maint: ref=harv (link)
• [Fanchon 2001] Fanchon, Jean-Louis (2001). Guide de mécanique : sciences et technologies industrielles (in French). Nathan. pp. 467–471. ISBN 978-2-09-178965-1.CS1 maint: ref=harv (link)
• [Fanchon 2011] Fanchon, Jean-Louis (2011). "Calcul des coussinets (régime non hydrodynamique)". Guide des sciences et technologies industrielles (in French). Afnor/Nathan. pp. 255–256. ISBN 978-2-09-161590-5.CS1 maint: ref=harv (link)
• [GCM 2000] Texeido, C.; Jouanne, J.-C.; Bauwe, B.; Chambraud, P.; Ignatio, G.; Guérin, C. (2000). Guide de construction mécanique (in French). Delagrave. pp. 110–116, 176–180. ISBN 978-2-206-08224-0.
• [SG 2003] Spenlé, D.; Gourhant, R. (2003). Guide du calcul en mécanique : maîtriser la performance des systèmes industriels (in French). Hachette technique. pp. 139–140. ISBN 2-01-16-8835-3.