Objective
To determine the stress-strain state of a cantilever frame subjected to a concentrated load.
Reference
A. Campa, R. Chappert та R. Picand, La mecanique par les problemas, fasc. 4: Resistance des materiaux, Paris, Foucher, 1987.
Problem statement
To determine the vertical displacements Z at the nodes where the horizontal bars are connected to the vertical bar (points B and D), as well as the bending moments My, shear forces Qz, and axial forces Nx at the fixed nodes of the horizontal bars (points A and C).
Design model
The cantilever frame consists of two horizontal bars of equal length L, fixed at one end (points A and C) and connected by a vertical bar of length l at the other ends (points B and D).
The horizontal bars have high axial stiffness, while the vertical bar has high axial stiffness as well as bending stiffness.
A vertical concentrated load F is applied at the node where the lower horizontal bar and the vertical bar are connected (point D).
Geometry
Length of horizontal bars L = 2 m
Length of vertical bar l = 0,2 m
Moment of inertia of the cross-section of horizontal bars Iz = (4/3) * 10-8 m4
Material properties
Modulus of elasticity Е = 2,0 * 1011 Pa
Loads
Vertical concentrated load: F = 1 kN
Note
The design model is a 2D frame consisting of three bar elements of FE type 10.
The boundary conditions are provided by applying restraints in the X, Z, and uY degrees of freedom (points A and C).
The axial stiffness (E*A) of the horizontal and vertical bars is taken as 1.0 × 10¹² N, while the bending stiffness (E·I) of the vertical bar is taken as 1.0 × 10¹² N·m².
Nodes: 4.
Output data
Comparison of calculation results
| Parameters | Analytical solution | LIRA-FEM | Error, % |
| Vertical displacement Z (point B), m | -0,125 | -0,125 | 0 |
| Vertical displacement Z (point D), m | -0,125 | -0,125 | 0 |
| Bending moment My (point A), N*m | -500 | -500 | 0 |
| Bending moment My (point C), N*m | -500 | -500 | 0 |
| Shear force Qz (point A), N | 500 | 500 | 0 |
| Shear force Qz (point C), N | 500 | 500 | 0 |
| Axial force Nx (point A), N | 5000 | 5000 | 0 |
| Axial force Nx (point C), N | -5000 | -5000 | 0 |
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