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Circular Ring Analysis with equal radial forces Equations and Calculator

Circular Ring Moment, Hoop Load, and Radial Shear Equations and Calculator #7.
Ring under any number of equal radial forces equally spaced.

Per. Roarks Formulas for Stress and Strain Formulas for Circular Rings Section 9, Reference, loading, and load terms #7.
Formulas for moments, loads, and deformations and some selected numerical values.

Circular Ring Loading #7	Circular Ring Dimensional Properties
Resultant moment, hoop load, and radial shear

Preview: Circular Ring Moment, Hoop Load, and Radial Shear Calculator #7

Formulas for moment, hoop load, radial shear and deformations Ring under any number of equal radial forces equally spaced.

For 0 < x < θ

Moment
M = [ W R ( u/s - k₂ / θ ) ]

Hoop Stress
N = W u / ( 2 s )

Radial Shear
V = - ( - W z ) / ( 2 s)

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 / AR² for thin rings
A = π [ R² - R_i² ]

Transverse (radial) shear deformation factor β

β = 2F (1 + v) e / R for thick rings
β = FEI / GAR² for thin rings

k₁ = 1 - α + β

k₂ = 1 - α

Plastic section modulus
F = Z / I c
Z = Z_x = Z_y = 1.33R³
I_x = I_y = ( π / 4 ) ( R⁴ - R_i⁴)

 

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 (length⁴). [Note that for a pipe or cylinder, a representative segment of unit axial length may be used by replacing EI by Et³ / 12 (1 - v² )
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.

Supplemental formulas (not included in calculator)

At each load position

Radial displacement at each load point = ΔR_B

Radial displacement at x = 0.20, ... = ΔR_A

Reference:

Roarks Formulas for Stress and Strain, 7th Edition, Table 9.2 Reference No. 5, loading, and load terms.