Stability of the Frame of Two Simply Supported Equally Loaded Rigid Columns Rigidly Interconnected by a Girder
Objective: Determination of the critical value of the concentrated longitudinal forces of the same value acting on two simply supported equally loaded rigid columns of the frame rigidly interconnected by a girder corresponding to the moment of buckling of the frame.
Initial data file: Frame_leg_hard.rar
Problem formulation: Two simply supported rigid columns of the frame rigidly interconnected by a girder are subjected to the action of concentrated longitudinal forces of the same value N. The axial stiffness of the girder is assumed to be significant in order to exclude its effect on the solution of the problem. Determine the critical value of the concentrated longitudinal forces Ncr, corresponding to the moment of buckling of the frame.
References: A. V. Perelmuter, V. I. Slivker, Handbook of Mechanical Stability in Engineering. Volume 2. Stability of Elastically Deformable Mechanical Systems, Moscow, SACD SOFT, 2010, p. 173.
Initial data:
| L = 10.0 m | - length of the girder of the frame; |
| H = 6.0 m | - height of the columns of the frame; |
| EA = 1.0·108 t | - axial stiffness of the girder; |
| EI = 1.0·104 t∙m2 | - bending stiffness of the girder; |
| N = 1.0·103 t | - initial value of the concentrated longitudinal forces on the columns of the frame. |
Finite element model: Design model – plane frame, columns – 2 elements of type 100 (two-node rigid bodies with the constraints in the directions X, Z, UY, support master nodes and slave nodes on the adjacent girder), girder – 10 elements of type 2 (the spacing of the finite element mesh along the longitudinal axes is 1.0 m). Boundary conditions are provided by imposing constraints on the support nodes of the columns in the directions of the degrees of freedom X and Z. The action with the initial value of the concentrated longitudinal forces N is specified in the beam-to-column joints. Number of nodes in the design model – 13.
Results in SCAD
Design model
Buckling mode
Comparison of solutions:
|
Parameter |
Theory |
SCAD |
Deviation, % |
|---|---|---|---|
|
Critical value of the concentrated longitudinal forces Ncr, t |
1000 |
0.999975∙1000 = = 1000 |
0.00 |
Notes: In the exact analytical solution the critical value of the concentrated longitudinal forces Ncr, corresponding to the moment of buckling of the frame can be determined according to the following formula:
\[ N_{cr} =\frac{6\cdot EI}{L\cdot H}. \]