Stability of the System of Three Equally Loaded Columns of Different Rigidity Hingedly Interconnected by Girders
Objective: Determination of the critical value of the concentrated longitudinal forces of the same value acting on the system of three columns of different rigidity hingedly interconnected by girders corresponding to the moment of its buckling. Determination of the unsupported lengths of the columns.
Initial data file: Frame_5a1.spr
Problem formulation: Three columns of different rigidity embedded into the foundation and hingedly interconnected into a system by girders are subjected to the action of concentrated longitudinal forces of the same value N. The axial stiffness values of the girders and columns are assumed to be significant in order to exclude their 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 system. Determine the unsupported lengths of the columns H0.
References: N. P. Melnikov, V. M. Vakhurkin, B. G. Lozhkin, Stability Analysis of Bar Systems. Reference data and examples, Moscow, Design Institute of Steel Structures, Issue 1395, 1954, p. 34.
Initial data:
| L = 5.0 m | - length of the girders of the frame; |
| H = 7.5 m | - height of the columns of the frame; |
| EA = 1.0·109 kN | - axial stiffness of the columns; |
| EIС1 = 1.14·104 kN∙m2 | - bending stiffness of the left column; |
| EIС2 = 2.28·105 kN∙m2 | - bending stiffness of the middle column; |
| EIС3 = 4.56·105 kN∙m2 | - bending stiffness of the right column; |
| N = 1.0·103 kN | - initial value of the concentrated longitudinal forces on the columns of the system. |
Finite element model: Design model – plane frame, columns – 45 elements of type 2 (the spacing of the finite element mesh along the longitudinal axes is 0.5 m), girders – 2 elements of type 100 (three-node rigid bodies with the constraints in the directions X and Z, master nodes in the middle of the girder spans, and slave nodes on the connected columns). Boundary conditions are provided by imposing constraints on the support nodes of the columns in the directions of the degrees of freedom X, Z, UY. 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 – 50.
Results in SCAD
Design model
Buckling mode
Comparison of solutions:
|
Parameter |
Theory |
SCAD |
Deviation, % |
|---|---|---|---|
|
Critical value of the concentrated longitudinal forces Ncr, kN |
1159.2 (1166.8) |
1.1591∙1000 = = 1159.1 |
0.01 (0.66) |
|
Unsupported length of the left column (С1) H0, m |
9.8522 (9.8198) |
9.8523 |
0.00 (0.33) |
|
Unsupported length of the middle column (С2) H0, m |
13.9331 (13.8873) |
13.9332 |
0.00 (0.33) |
|
Unsupported length of the right column (С3) H0, m |
19.7043 (19.6396) |
19.7042 |
0.00 (0.33) |
The values of the approximate solution by the equivalent frame method are given in brackets.
Notes: In the exact analytical solution the critical value of the concentrated longitudinal forces Ncr, corresponding to the moment of buckling of the system, and the unsupported lengths of the columns H0 can be determined according to the following formulas:
\[ N_{cr} =\nu^{2}\cdot \frac{EI_{С1} }{H^{2}}, \]
where ν (critical load parameter) is determined by solving the transcendental equation:
\[ {\begin{array}{*{20}c} {\left( {tg\left( \nu \right)-\nu } \right)\cdot \left( {tg\left( {\frac{\sqrt 2 }{2}\cdot \nu } \right)-\frac{\sqrt 2 }{2}\cdot \nu } \right)+\sqrt 2 \cdot \left( {tg\left( \nu \right)-\nu } \right)\cdot \left( {tg\left( {\frac{1}{2}\cdot \nu } \right)-\frac{1}{2}\cdot \nu } \right)+} \\ {2\cdot \left( {tg\left( {\frac{\sqrt 2 }{2}\cdot \nu } \right)-\frac{\sqrt 2 }{2}\cdot \nu } \right)\cdot \left( {tg\left( {\frac{1}{2}\cdot \nu } \right)-\frac{1}{2}\cdot \nu } \right)=0;} \\ \end{array} } \] \[ {\begin{array}{*{20}c} {С1:} & {H_{0} =\frac{\pi \cdot H}{\nu };} & {С2:} & {H_{0} =\sqrt {2\cdot } \frac{\pi \cdot H}{\nu };} & {С3:} & {H_{0} =2\cdot \frac{\pi \cdot H}{\nu }.} \\ \end{array} } \]
In the approximate analytical solution the critical value of the concentrated longitudinal forces Ncr, corresponding to the moment of buckling of the system, and the unsupported lengths of the columns H0 can be determined according to the following formulas:
\[ N_{cr} =\frac{7}{3}\cdot \frac{\pi^{2}\cdot EI_{С1} }{\left( {2\cdot H} \right)^{2}}; \] \[ {\begin{array}{*{20}c} {С1:} & {H_{0} =\pi \cdot \sqrt {\frac{EI_{С1} }{N_{cr} }} ;} & {С2:} & {H_{0} =\sqrt {2\cdot } \sqrt {\frac{EI_{С1} }{N_{cr} }} ;} & {С3:} & {H_{0} =2\cdot \sqrt {\frac{EI_{С1} }{N_{cr} }} .} \\ \end{array} } \]