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Circular Ring Analysis No 14 Roarks Formulas for Stress and Strain Equations and Calculator

Circular Ring Moment, Hoop Load, and Radial Shear Equations and Calculator #14.

Per. Roarks Formulas for Stress and Strain Formulas for Circular Rings Section 9, Reference, loading, and load terms #14.
Formulas for moments, loads, and deformations and some selected numerical values.

Circular Ring Loading #14	Circular Ring Dimensional Properties
Resultant moment, hoop load, and radial shear

Preview: Circular Ring Moment, Hoop Load, and Radial Shear Calculator #14

The radial pressure w_x vaies with (x - θ)² from 0 at x = θ to w at x = π

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

θ ≥ π / 2

Moment Formula

Hoop Stress Formula

Radial Shear Fomula

V_A = 0

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 supported in load terms equations.

Unit step function defined by use of ⟨ ⟩, See explanation at bottom of page.
⟨ x - θ ⟩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 / 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)

Deformation in the Horizontal Axis (Click for Larger View)

Deformation in the Vertical Axis (Click on imge for larger view)

Change of Length of Cylinder (Click for larger view)

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.

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. 14, loading, and load terms.