Bearing pressure explained

Bearing pressure is a particular case of contact mechanics 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. This problem is also referred to as bearing resistance.^[1]

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 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.

Case of 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

\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

In this first modeling, the pressure is uniform. It is equal to:

P = (F)/(D x L) = (radial load)/(projected area) .

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:^[3]

P(θ) = P_max⋅cos θwith

P _{max} = (4)/(\pi) ⋅ (F)/(L D) .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:

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

P _{max} = (4 F)/(L D) x (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)).

Notes and References

1. EN 1993-1-8:2005 Eurocode 3: Design of steel structures - Part 1-8: Design of joints
2. due to the clearance, the diameter of the bore is bigger than the diameter of the male cylinder; however, we suppose that the diameters are close to each othert
3. Book: Budynas, Richard G. . Shigley's mechanical engineering design . McGraw-Hill . J. Keith . Nisbett . Joseph Edward . Shigley . 2011 . 978-0-07-352928-8 . 9th . New York . 664, eq. 12-31 . 436031178.