LIRALAND

Test 1.13 Curved Beam (Golovin’s Problem)

Updated Feb 06, 2026 | Published Oct 19, 2025

Objective

To determine the stress state of a curved narrow beam with a rectangular cross-section under bending by a concentrated force applied to the free end.

Reference

Demidov S.P. Theory of Elasticity, Moscow, Vysshaya Shkola, 1979.

Problem statement

To determine the components of the stress tensor σrr and σθθ in polar coordinates for beam cross-section l–l, located at an angle θ = 90° from the end face of the free end.

Design model

A force P is applied to the free end of a cantilever beam of unit thickness. The force acts parallel to the end face of the beam in the plane of the circular axis of the beam.

Initial geometry of analytical model

Initial geometry of analytical model

Initial geometry of FE model, variant 1

Initial geometry of FE model, variant 2

Variant 1

Âàð³àíò 2


Initial geometry of FE model

Geometry

Beam thickness h = 1 m
Inner radius r1 = 5 m
Outer radius r2 = 15 m

Material properties

Modulus of elasticity Å = 3 * 107 kPa
Poisson's ratio ν = 0,2

Boundary conditions

Restraints in all degrees of freedom on face m-m.

Loads

Load distributed over face n-n, r = 5 kN

Output data

Design and deformed shapes, variant 1

Design and deformed shapes, variant 2

Variant 1

Variant 2


Design and deformed models

Stress mosaic plot σrr (Ny), kN/m2, variant 1

Stress mosaic plot σrr (Ny), kN/m2, variant 2

Variant 1

Variant 2


Stress mosaic plot σrr (Ny), kN/m2

Stress mosaic plot σθθ (Nõ), kN/m2, variant 1

Stress mosaic plot σθθ (Nõ), kN/m2, variant 2

Variant 1

Variant 2


Stress mosaic plot σθθ (Nõ), kN/m2

Stress diagram σrr (Ny) along section l-l, kN/m2, variant 1

Stress diagram σrr (Ny) along section l-l, kN/m2, variant 2

Variant 1

Variant 2


Stress diagram σrr (Ny) along section l-l, kN/m2

Stress diagram σθθ (Nõ) along section l-l, kN/m2, variant 1

Stress diagram σθθ (Nõ) along section l-l, kN/m2, variant 2

Variant 1

Âàð³àíò 2


Stress diagram σθθ (Nõ) along section l-l, kN/m2

Comparison of calculation results

Without additional side nodes:

The unknown Point in the section l-l Analytical solution LIRA-FEM Error, %
Variant
1 		Variant
2 		Variant
1 		Variant
2
Stress σrr, kN/m2 r = 5m 0 -0,1598 -0,0844 ≈0 ≈0
r = 7,415m -0,8375 -0,8309 -0,8351 0,7881 0,2866
r = 15m 0 -0,0199 -0,0195 ≈0 ≈0
Stress σθθ, kN/m2 r = 5m -5,3581 -4,9143 -5,1374 8,2828 4,119
r = 15m 1,7860 1,7860 1,7359 5,7671 2,8052

With additional side nodes:

The unknown Point in the section l-l Analytical solution LIRA-FEM Error, %
Variant
1 		Variant
2 		Variant 
1 		Variant 
2
Stress σrr, kN/m2 r = 5m 0 -0,1583 -0,084 ≈0 ≈0
r = 7,415m -0,8375 -0,8319 -0,8354 0,6687 0,2507
r = 15m 0 -0,0201 -0,01 ≈0 ≈0
Stress σθθ, kN/m2 r = 5m -5,3581 -4,9278 -5,1409 8,0308 4,0537
r = 15m 1,7860 1,7324 1,7595 3,0011 1,4838

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