Bearing pressure

There are two ways to obtain this result.

Hemicylindrical body at the equilibrium in a fluid with hydrostatic pressure.

First, we can consider a hemicylinder in a fluid, with a uniform hydrostatic pressure. The equilibrium is achieved when the resulting force on the flat surface is equal to the resulting force on the curved one. The flat surface is a D × L rectangle, therefore

F = P × (D × L)

q.e.d.

The elementary force dF, due to the pressure on a surface element dS, has two components: dF_x and dF_y.

Second, we can integrate the pressure elementary forces. Consider a small surface dS on the cylindrical part, parallel to a generating line; its length is L, and it is bound by the angles θ and θ + dθ. This small surface element can be considered as a flat rectangle which dimensions are L × (dθ × D/2). The pressure force on the surface is equal to

dF = P × dS = ½ × P × D × L × dθ

The (y, z) plane is a plane of reflection symmetry, so the x compound of this force is annihilated by the force on the symmetrical surface element. The y compound of this force is equal to:

dF_y = cos(θ) dF = ½ × cos(θ) × P × D × L × dθ.

The resulting force is equal to

${\displaystyle F_{y}=\int _{-\pi /2}^{\pi /2}{\frac {1}{2}}\times P\times D\times L\times \cos(\theta )\times \mathrm {d} \theta ={\frac {1}{2}}\times P\times D\times L\times \left[\sin(\theta )\right]_{-\pi /2}^{\pi /2}=P\times D\times L}$

q.e.d.

This calculation is similar to the case of a cylindrical vessel under pressure.

Elastic deformation in case of a male-female cylinders contact.

Relationship between pressure, clearance and contact angle

The part no. 1 is the containing cylinder (female, concave), the part no. 2 is the contained cylinder (male, convex); the center of the cylinder i is O_i, and its radius is R_i.

The reference position is an ideal situation where both cylinders are concentric. The clearing, expressed as a radius (not diameter), is:

j = R₁ - R₂.

Under the load, the part 2 gets in contact with the part 1, the he surfaces deform. we suppose that the cylinder 2 is rigid (no deformation), and that the cylinder 1 is an elastic body. The indentation of 2 into 1 has a depth of δ_max; the cylinder movement is e (excentration):

e = O₁O₂ = j + δ_max.

We considere the frame at the center of the cylinder 1 (O₁, x, y). Let M be a point on the contact surface; θ is the angle (-y, O₁M). The displacement of the surface, δ, is:

δ(θ) = O₁M - R₁.

with δ(0) = δ_max. The coordinates of M are:

M((R₁ + δ(θ)⋅sin θ) ; -(R₁ + δ(θ))⋅cos θ)

and the coordinates of O₂:

O₂(0 ; -e).

Consider the frame (O₁, u, v), where the axis u is (O₁M). In this frame, the coordinates are:

M(R₁ + δ(θ) ; 0)

O₂(e⋅cos θ ; -e⋅sin θ)

We know that

${\displaystyle {\overrightarrow {O_{1}M}}={\overrightarrow {O_{1}O_{2}}}+{\overrightarrow {O_{2}M}}}$

thus

${\displaystyle \left\{{\begin{aligned}R_{1}+\delta (\theta )&=&e\cdot \cos \theta +R_{2}\cos \varphi \\0&=&-e\cdot \sin \theta +R_{2}\sin \varphi \end{aligned}}\right.}$

then we use the expression of e and R₁ = j + R₂:

${\displaystyle \left\{{\begin{aligned}j+R_{2}+\delta (\theta )&=&(j+\delta _{\max })\cdot \cos \theta +R_{2}\cos \varphi \\0&=&-(j+\delta _{\max })\cdot \sin \theta +R_{2}\sin \varphi \end{aligned}}\right.}$

The deformations are small, as we are in the elastic domain. Thus, δ_max ≪ R₁ and therefore |φ| ≪ 1, i.e.

cos φ ≃ 1

sin φ ≃ φ (in radians)

thus

${\displaystyle \left\{{\begin{aligned}(j+\delta _{\max })\cdot \cos \theta +R_{2}-(j+R_{2}+\delta (\theta ))&=&0\\(j+\delta _{\max })\cdot \sin \theta -R_{2}\cdot \varphi &=&0\end{aligned}}\right.}$

and

${\displaystyle \left\{{\begin{aligned}(j+\delta _{\max })\cdot \cos \theta -j-\delta (\theta )&=&0\\(j+\delta _{\max })\cdot \sin \theta -R_{2}\cdot \varphi &=&0\end{aligned}}\right.}$

At θ = θ₀, δ(0) = 0 and the first equation is

${\displaystyle (j+\delta _{\max })\cdot \cos \theta _{0}-j=0\Rightarrow \delta _{\max }={\frac {j(1-\cos \theta _{0})}{\cos \theta _{0}}}\Leftrightarrow \cos \theta _{0}={\frac {j}{j+\delta _{\max }}}}$

and thus

${\displaystyle \left(j+{\frac {j(1-\cos \theta _{0})}{\cos \theta _{0}}}\right)\cdot \cos \theta -j-\delta (\theta )=0}$

${\displaystyle \Longrightarrow \delta (\theta )=j\left({\frac {\cos \theta }{\cos \theta _{0}}}-1\right)}$ [1].

If we use the law of elasticity for a metal (α = 1):

${\displaystyle \left\{{\begin{aligned}P(\theta )&=&K\cdot j\left({\frac {\cos \theta }{\cos \theta _{0}}}-1\right)\\P_{\max }&=&K\cdot j\cdot {\frac {1-\cos \theta _{0}}{\cos \theta _{0}}}\end{aligned}}\right.}$ [2]

The pressure is an affine function of cos θ:

P(θ) = A⋅cos θ - B

with A = K⋅j/cos θ₀ and B = A⋅cos θ₀.

Case where the clearance can be neglected

If j ≃ 0 (R₁ ≃ R₂), then the contact is on the whole half-perimeter: 2θ₀ ≃ π and cos θ₀ ≃ 0. The value of 1/cos θ₀ rise towards infinity, thus

${\displaystyle \delta (\theta )\simeq j{\frac {\cos \theta }{\cos \theta _{0}}}}$

As j and cos θ₀ both tend towards 0, the ratio j/cos θ₀ is not defined when j goes to 0. In mechanical engineering, j = 0 is an uncertain fit, it is a nonsense, both mathematically and mechanically. We are looking for a limit function

${\displaystyle P_{\pi /2}(\theta )=\lim _{\theta _{0}\to \pi /2}P_{\theta _{0}}(\theta )}$.

So, the pressure is a sinusoid function of θ:

${\displaystyle P(\theta )=K\cdot {\frac {j}{\cos \theta _{0}}}\cdot \cos \theta }$

thus

P(θ) = P_max⋅cos θ

with

${\displaystyle P_{\max }=K\cdot {\frac {j}{\cos \theta _{0}}}}$.

Consider an infinitesimal element of surface dS bound by θ and θ + dθ. As in the case of the uniform pressure, we have

dF_y(θ) = cos(θ)dF = ½ × cos(θ) × P(θ) × D × L × dθ = ½ × cos²(θ) × P_max × D × L × dθ.

When we integrate between -π/2 and π/2, the result is:

${\displaystyle F=\int _{-\pi /2}^{\pi /2}\mathrm {d} F_{y}(\theta )\mathrm {d} \theta ={\frac {1}{2}}P_{\max }DL\int _{-\pi /2}^{\pi /2}\cos ^{2}\theta \mathrm {d} \theta }$

We know that (e.g. using the Euler's formula):

${\displaystyle \int _{-\pi /2}^{\pi /2}\cos ^{2}\theta \mathrm {d} \theta ={\frac {1}{4}}\left[2\theta +\sin 2\theta \right]_{-\pi /2}^{\pi /2}={\frac {1}{2}}\left[\theta +\sin \theta \cos \theta \right]_{-\pi /2}^{\pi /2}={\frac {\pi }{2}}}$

therefore

${\displaystyle F={\frac {\pi }{4}}P_{\max }DL}$

and thus

${\displaystyle P_{\max }={\frac {4}{\pi }}{\frac {F}{DL}}}$

q.e.d.

Case where the clearance can not be neglected

The force on an infinitesimal element of surface is:

dF(θ) = P(θ)dS = Kδ(θ)dS = ½ × K × j × cos θ/cos θ₀ - 1) × dS

${\displaystyle \mathrm {d} F_{y}={\frac {Kj}{2}}\left({\frac {\cos \theta }{\cos \theta _{0}}}-1\right)DL\cos \theta \mathrm {d} \theta }$

${\displaystyle F={\frac {KjDL}{2}}\int _{-\theta _{0}}^{\theta _{0}}\left({\frac {\cos ^{2}\theta }{\cos \theta _{0}}}-\cos \theta \right)\mathrm {d} \theta ={\frac {KjDL}{2}}\left[{\frac {\theta +\sin \theta \cos \theta }{2\cos \theta _{0}}}-\sin \theta \right]_{-\theta _{0}}^{\theta _{0}}}$

thus

${\displaystyle F={\frac {KjDL}{2}}\left({\frac {2\theta _{0}+2\sin \theta _{0}\cos \theta _{0}}{2\cos \theta _{0}}}-2\sin \theta _{0}\right)={\frac {KjDL}{2}}\left({\frac {\theta _{0}-\sin \theta _{0}\cos \theta _{0}}{\cos \theta _{0}}}\right)}$.

We recognise the trigonometric identity sin 2θ = 2 sin θ cos θ :

${\displaystyle F={\frac {KjDL}{4\cos \theta _{0}}}(2\theta _{0}-\sin 2\theta _{0})}$

thus

${\displaystyle K={\frac {4F\cos \theta _{0}}{jDL(2\theta _{0}-\sin 2\theta _{0})}}}$

and therefore:

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

q.e.d.