Parameters
Load Case
Ring Geometry
Mean radius R500 mm
Section width b50 mm
Section height h50 mm
Material & Load
Elastic modulus E200 GPa
Yield stress σ_y250 MPa
Load intensity P / q10.0 kN
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M_max [kN·m]
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N_max [kN]
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Max Stress [MPa]
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Max Deflection [mm]
Bending Moment Distribution (full ring)
Normal Force Distribution
Governing Equations
Diametral point loads (angle θ):
$$M(\theta) = \frac{PR}{\pi}\left(1 - \frac{\pi}{2}|\sin\theta|\right), \quad 0 \le \theta \le \pi$$ $$N(\theta) = -\frac{P}{2}\cos\theta - \frac{P}{\pi}|\sin\theta|$$Internal pressure (uniform radial load q):
$$N = qR \text{ (hoop force)},\quad M = 0,\quad \sigma_\theta = \frac{qR}{t}$$Combined section stress (rectangular section): $\sigma = \dfrac{N}{A} \pm \dfrac{M \cdot h/2}{I}$
Diametral deflection under point loads:
$$\delta = \frac{PR^3}{EI}\left(\frac{\pi}{4} - \frac{2}{\pi}\right) \times 2$$
CAE Note: Ring theory underpins the analysis of pipe elbows, storage tank rings, and pipeline spans. ASME B31.3 uses the ring flexibility parameter to compute the Stress Intensification Factor (SIF) for curved pipe fittings.