Objective
To determine the deformed state of a semicircular arch of constant cross-section with hinged supports subjected to a concentrated load acting in its plane.
Reference
P. Dellus, Resistance de materiaux, Paris, Technique et Vulgarisation, 1958.
Problem statement
To determine the arch deflection along the Z-axis, the displacement X of the hinged movable support, and the rotation angles uY of the support hinges.
Design model
A semicircular arch of constant cross‑section, with a hinged fixed support and a hinged movable support at the springing level. The arch is subjected to a concentrated load F in its plane at the crown, directed normal to the longitudinal axis towards the springings.
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Initial geometry of analytical model
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Initial geometry of FE model
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Geometry
Radius of the arc of the longitudinal axis of the semicircular arch: r = 1,0 m
Outer diameter of the circular cross‑section of the arch: de = 0,020 m
Inner diameter of the circular cross‑section of the arch: di = 0,016 m
Material properties
Modulus of elasticity Е = 2,0 * 1011 Pa
Loads
Concentrated load F = 100 N
Note
Design model – plane frame that consists of 48 bar elements of FE type 10.
The boundary conditions are provided by applying restraints:
– in the X and Z degrees of freedom (DOF) directions for the hinged fixed support;
– in the Z degree of freedom (DOF) direction for the hinged movable support.
Nodes: 49.
Output data
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Design and deformed models
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Vertical displacements Z (m)
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Horizontal displacements X (m)
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Rotation angles uY (rad*1000)
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Comparison of calculation results
| Parameters | Analytical solution | LIRA-FEM | Error, % |
| Deflection of the arch along Z-axis, m | -0,0192 | -0,0192 | 0 |
| Displacement of the hinged movable support in X-axis, m | 0,053 | 0,053 | 0 |
| Rotation angle of the hinged movable support uY, rad×1000 | -30,774 | -30,79 | 0 |
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Rotation angle of the hinged fixed support uY, rad×1000 |
30,774 | 30,77 | 0 |
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