Objective
To determine the stress-strain state of a single-span hinged plane frame with a double-pitched girder subjected to a vertical uniformly distributed load, concentrated vertical and horizontal loads, and a bending moment.
Reference
JC Bianchi, Rapport de la Socotec, Paris, non publie, 1964
Problem statement
To determine the vertical displacement Z at the ridge node of the girder (point C), as well as the axial force N and shear force Q at the support node of the left column (point A).
Design model
A single‑span hinged frame with rigid fixation of the double‑pitched rafter to the columns is subjected to:
– a vertical uniformly distributed load Pzx over the length of the left half span of the rafter 0.5*L,
– a concentrated vertical force F1 at ridge node 2, and
– a bending moment M at the rafter-to-column connection.
|
|
|
|
Initial geometry of analytical model
|
Initial geometry of FE model
|
Geometry
Columns L1:
Height h = 8 m;
Axial stiffness EA1 = 1*1010 N;
Bending stiffness ЕI1 = 10,5*107 N m2;
Girder L2:
Span length L = 20 m;
Rise a = 4 m;
Length of the slope b = ((0.5∙20.0)2 + 42)0,5;
Axial stiffness EA2 = 1*1010 N;
Bending stiffness ЕI2 = 5,25*107 N m2
Material properties
Modulus of elasticity for bars in the model Е = 2,0 * 1011 Pa
Loads
Vertical uniformly distributed load over the length of the left half-span of the girder Pzx = 3*103 N/m;
The same load over the length of the left roof slope of the girder Px = 2,78543*103 N/m;
Concentrated vertical load applied at the ridge node of the girder F1 = 2*104 N;
Concentrated horizontal load at the girder‑to‑left-column connection F2 = 1*104 N;
Concentrated bending moment at the girder‑to‑left-column connection М = 1*105 N*m.
Note
Design model – plane frame, girder – 2 elements of FE type 2, columns – 2 elements of FE type 2.
The boundary conditions are provided by applying restraints in the X and Z degrees of freedom (DOF) directions at the hinged support nodes.
Number of nodes in design model – 5.
Output data
|
Design and deformed models
|
Values for vertical displacements Z (m)
|
|
Values of axial forces N (N)
|
Values for shear forces Q(N)
|
Comparison of calculation results
| Parameters | Analytical solution | LIRA-FEM | Error, % |
| Vertical displacement ZС, m | -0,03072 | -0,03075 | 0,1 |
| Axial force NА, N | -31,500 | -31,500 | 0 |
| Shear force QА, N | 20239,4 | 20239 | 0 |
Download verification test
If you find a mistake and want to inform us about it, select the mistake, then hold down the CTRL key and click ENTER.
Comments