Objective
To determine the stress-strain state of a system of intersecting bars subjected to a distributed load and a concentrated load acting in their plane.
Reference
S. Timoshenko et D.H. Young, Theorie des constructions, Paris, Librairie Polytechnique Beranger, 1949, p. 412-416.
Problem statement
To determine the rotation angle UY at the common joint of the intersecting bars (point A) and the bending moments M in the bars on both sides of the joint.
Design model
The system consists of two intersecting bars with square cross-sections: a horizontal bar (BD) and a vertical bar (CE), rigidly connected at the common joint (point A).
The horizontal bar is rigidly fixed at the left and right nodes (points D and B).
The vertical bar is rigidly fixed at the lower node (point E) and hinged at the upper node (point C).
A vertical concentrated load F is applied at the midpoint of the left span of the horizontal bar (point G). A vertical uniformly distributed load p acts on the right span of the horizontal bar (AB).
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Initial geometry of analytical model
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Initial geometry of FE model
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Geometry
Length of the left span of the horizontal bar: LАD = 1,0 m
Length of the cross-sectional side of the left span of the horizontal bar: bAD = 1,0 m
Length of the right span of the horizontal bar: LАB = 4,0 m
Length of the cross-sectional side of the right span of the horizontal bar: bAB = 4,0 m
Length of the upper segment of the vertical bar: LАС = 1,0 m
Length of cross-sectional side of the upper segment of the vertical bar: bAС = 1,0 m
Length of the lower segment of the vertical bar: LАЕ = 2,0 m
Length of cross-sectional side of the lower segment of the vertical bar: bAЕ = 2,0 m
Material properties
Modulus of elasticity for bars in the system Е = 2,0 * 1011 Pa
Loads
Vertical concentrated load: F = 100 kN
Vertical uniformly distributed load: p = 1,0 kN/m
Note
The design model is a 2D frame consisting of five bar elements of FE type 10.
The boundary conditions are provided by applying restraints:
- in the X and Z degrees of freedom at the hinged node (point C);
- in the X, Z, and UY degrees of freedom at the rigidly fixed nodes.
Nodes: 6.
Output data
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Design and deformed models of the truss
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Values of rotation angles UY (rad*1000)
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Values of bending moments M (N*m)
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Comparison of calculation results
| Parameters | Analytical solution | LIRA-FEM | Error, % |
| Rotation angle UY (point A), rad*1000 | -227,12 | -227,4 | 0,12 |
| Bending moment M (bar AD), N * m | -12348,6 | -12348 | 0,01 |
| Bending moment M (bar AВ), N * m | -11023,7 | -11021 | 0,02 |
| Bending moment M (bar AС), N * m | 113,6 | 113,71 | 0,09 |
| Bending moment M (bar AЕ), N * m | -1211,3 | -1212,8 | 0,12 |
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