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### Circular Ring Loading Analysis No. 12 Roarks Formulas for Stress and Strain Equations and Calculator

**Circular Ring Moment, Hoop Load, and Radial Shear Equations and Calculator #12**.

Per. Roarks Formulas for Stress and Strain Formulas for Circular Rings Section 9, Reference, loading, and load terms #12.

Formulas for moments, loads, and deformations and some selected numerical values.

Circular Ring Loading #12 |
Circular Ring Dimensional Properties |

Resultant moment, hoop load, and radial shear |

Preview: Circular Ring Moment, Hoop Load, and Radial Shear Calculator #12

General formulas for moment, hoop load, radial shear and deformations.

θ ≥ π / 2

Moment

M = M_{A} - N_{A} R ( 1 - u) + V_{A} R z + LT_{M}

Hoop Stress

N = N_{A} u + V_{a} z + LT_{N}

Radial Shear

V = - N_{A} z + V_{A} u + LT_{v}

LT_{M} LT_{N}, and LT_{V} are load terms for several types of load.

Note: Loads beyond 180° not support in load terms equations.

LT_{M} = - w R^{2} [ 1 - cos(x - θ ) ] ⟨ x - θ ⟩^{0}

LT_{N} = - w R [ 1 - cos(x - θ ) ] ⟨ x - θ ⟩^{0}

LT_{v} = - w R sin (x - θ ) ⟨ x - θ ⟩^{0}

Unit step function defined by use of ⟨ ⟩, See explanation at bottom of page.

⟨ x - θ ⟩^{0}

V_{A} = 0

R_{i} / t ≥ 10 means thin wall for pressure vessels (per. ASME Pressure Vessel Code)

Hoop Stress Deformation Factor α

α = e / R for thick rings

α = I / AR^{2} for thin rings

A = π [ R^{2} - R_{i}^{2} ]

Transverse (radial) shear deformation factor β

β = 2F (1 + v) e / R for thick rings

β = FEI / GAR^{2} for thin rings

k_{1} = 1 - α + β

k_{2} = 1 - α

Plastic section modulus

F = Z / I c

Z = Z_{x} = Z_{y} = 1.33R^{3}

I_{x} = I_{y} = ( π / 4 ) ( R^{4} - R_{i}^{4})

Deformation in the Horizontal Axis

Deformation in the Vertical Axis

Change of length of Circular Ring

Where (when used in equations and this calculator):

W = load (force);

v and w = unit loads (force per unit of circumferential length);

G = Shear modulus of elasticity

F = Shape factor for the cross section (= Z / I c)

Z = Plastic section modulus

w and v = unit loads (force per unit of circumferential length);

ρ = unit weight of contained liquid (force per unit volume);

M_{o} = applied couple (force-length);

M_{A}, M_{B}, M_{c}, and M are internal moments at A;B;C, and x, respectively, positive as shown.

N_{A} N, V_{A}, and V are internal forces, positive as shown.

E = modulus of elasticity (force per unit area);

*v* = Poisson’s ratio;

A = cross-sectional area (length squared);

R = Outside radius to the centroid of the cross section (length);

R_{i} = Inside radius to the centroid of the cross section;

t = wall thickness

I = area moment of inertia of ring cross section about the principal axis perpendicular to the plane of the ring (length^{4}). [Note that for a pipe or cylinder, a representative segment of unit axial length may be used by replacing EI by Et^{3} / 12 (1 - v^{2} )

e ≈ I / ( R A) positive distance measured radially inward from the centroidal axis of the cross section to the neutral axis of pure bending

θ, x, and Φ are angles (radians) and are limited to the range zero to π for all cases except 18 and 19

s = sin θ

c = cos θ

z = sin x,

u = cos x

n = sin Φ

m = cos Φ

α - Hoop stress deformation factor

ΔD_{V} = Change in vertical diameter (in, mm),

ΔD_{H} = Change in horizontal diameter (in, mm),

ΔL = Change in lower half of vertical diameterthe vertical motion realative to point C of a line connecting points B and D on the ring,

ΔL_{W} = Vertical motion relative to point C of a horizontal line connecting the load points on the ring,

ΔL_{WH} = Change in length of a horizontal line connecting the load points on the ring,

ψ = angular rotation (radians) of the load point in the plane of the ring and is positive in the direction of positive θ.

Unit step function defined by use of ⟨ ⟩

⟨ x - θ ⟩^{0}

if x < 0, ⟨ x - θ ⟩^{0} =0;

if x > 0, ⟨ x - θ ⟩^{0} =1;

At x = θ the unit step function is undefined just as vertical shear is undefined directly beneath a concentrated load. The use of the angle brackets ⟨ ⟩ is extended to other cases involving powers of the unit step function and the ordinary function ⟨ x - a ⟩^{n}

Thus, the quantity ⟨ x - a ⟩^{n} ⟨ x - a ⟩^{0} is shortened to ⟨ x - a ⟩^{n} and again is given a value of zero if x < a and is ( x-a )^{n} if x > a.

Supplemental formulas (not included in calculator)

Reference:

Roarks Formulas for Stress and Strain, 7th Edition, Table 9.2 Reference No. 11, loading, and load terms.