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2026-02-20

VE0086 | Curved Beam with Out-of-Plane Loading

Description

A quarter-circle beam with a rectangular cross-section w × h is loaded by means of an out-of-plane force F. While neglecting self-weight, the goal is to determine the total deflection u_z of the curved beam.

Material	Isotropic Linear Elastic	Modulus of Elasticity	E	210000.000	MPa
Material	Isotropic Linear Elastic	Poisson's Ratio	ν	0.296	-
Geometry	Rectangular Section	Radius	r	1.000	m
		Cross-Section Width	w	25.000	mm
		Cross-Section Height	h	50.000	mm
Load	Out-of-Plane	Force	F	1.000	kN

Curved Beam with Out-of-Plane Loading

Analytical Solution

The curved beam is loaded by a bending moment M_b, torsional moment M_t, and by a transverse force T. Considering the following schema, these loads at an arbitrary section are equal to:

M_{b} = Fa = Frsinφ

M_{t} = Fb = Fr (1 - cosφ)

T = F

Loads Acting on Curved Beam

Curved Beam with Out-of-Plane Loading - Schema

The deflection of the structure is determined according to Castigliano's second theorem:

u_{z} = \frac{d U}{d F} = \frac{d (U_{b} + U_{t} + U_{s})}{d F},

Castigliano's Second Theorem

Where the total strain energy U is composed of bending (U_b), torsional (U_t), and shear (U_s) components. Using polar coordinates (ds = r dφ):

U_{b} = \int_{L} \frac{M_{b}^{2} (s)}{2 E I_{y}} d s = \int_{0}^{π / 2} \frac{M_{b}^{2} (φ)}{2 E I_{y}} r d φ

U_{t} = \int_{L} \frac{M_{t}^{2} (s)}{2 G J} d s = \int_{0}^{π / 2} \frac{M_{t}^{2} (φ)}{2 G J} r d φ

U_{s} = \frac{6}{5} \int_{L} \frac{T^{2}}{2 G A} d s = \frac{6}{5} \int_{0}^{π / 2} \frac{T^{2}}{2 G A} r d φ

Strain Energy

The total deflection u_z is then equal to:

u_{z} = \frac{3 π F r^{3}}{E w h^{3}} + \frac{1.555 F r^{3}}{G h w^{3}} + \frac{3 π F r}{5 G h w} \approx 38.960 mm

Total Deflection

RFEM Settings

Modeled in RFEM 6.13 and RFEM 5.39
Element size: l_FE = 0.010 m
Isotropic linear elastic material
Mindlin plate bending theory

Results

Entity	Theory u_z [mm]	RFEM 6 u_z [mm]	Ratio [-]	RFEM 5 u_z [mm]	Ratio [-]
Member	38.960	38.973	1.000	38.973	1.000
Plate, horizontal		39.129	1.004	38.642	0.992
Plate, vertical		38.158	0.979	38.117	0.978
Solid		38.703	0.993	38.398	0.986

Model 000184 | Verification Example 000086 | 1

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Verification Example 000086 | 1

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