Rigid Body Restrained by Five Springs of the Same Rigidity Working Only in Tension Subjected to a Concentrated Force
Objective: Determination of the reactions of springs of the same rigidity working only in tension and restraining a rigid body from the action of a concentrated force applied to it in the structurally nonlinear formulation.
Initial data file: Contact_2.spr
Problem formulation: The rigid body in the shape of a square with the sides parallel to the coordinate axes is restrained at the corners by five springs of the same rigidity working only in tension as follows:
two springs (1 and 5) are installed in the lower left corner of the square, the angles between their longitudinal axes and the lower side of the square are 150° and 30° respectively;
one spring is installed in the upper left corner of the square (2), the angle between its longitudinal axis and the upper side of the square is 30°;
springs (4 and 3) are installed in the lower right and in the upper right corners of the square, the angle between their longitudinal axes and the lower and upper sides of the square respectively is 90°.
The concentrated force P is applied perpendicular to the middle of the left side of the square of the rigid body.
Determine the reactions in the springs Ri.
References: A.V. Perelmuter, V.I. Slivker, Design Models of Structures and a Possibility of Their Analysis, Moscow, SCAD SOFT, 2011, p. 147
Initial data:
| L = 20 м | - side of the square of the rigid body; |
| α1 = 150° | - angle between the axis of the spring 1 and the lower side of the square; |
| α2 = 30° | - angle between the axis of the spring 2 and the upper side of the square; |
| α3 = 90° | - angle between the axis of the spring 3 and the upper side of the square; |
| α4 = 90° | - angle between the axis of the spring 4 and the lower side of the square; |
| α5 = 30° | - angle between the axis of the spring 5 and the lower side of the square; |
| k = 1.00·106 kN/m | - axial stiffness of the springs; |
| P = 10.0 kN | - value of the concentrated force acting perpendicular to the middle of the left side of the square. |
Finite element model: Design model – plane frame. Element of the rigid body – 1 3D six-node rigid body element of type 100 (one master node lying at the intersection of the diagonals of the square, four slave nodes lying at the corners of the square, one slave node lying in the middle of the left side of the square). Elements of the springs – 5 two-node elements of unilateral constraints of type 352. Boundary conditions are provided by imposing constraints on the support nodes of the springs in the directions of the degrees of freedom X, Z. An element of the constraint of finite rigidity (type 51) of small value 0.001 kN/m in the direction of the X axis of the global coordinate system is introduced in the master node of the rigid body to provide the dimensional stability of the system during the nonlinear calculation. The results of the calculation are correct if there are no reactions in this constraint. The action is specified as a nodal load P (in the direction of the X axis of the global coordinate system). The nonlinear loading was generated for the incremental-iterative method with a loading factor - 1, number of steps - 1, number of iterations - 10 for the linear loading P. Number of nodes in the design model – 17.
Results in SCAD
Design model
Values of reactions in the support nodes of the springs along the X axis of the global coordinate system Rx, kN
Values of reactions in the support nodes of the springs along the Z axis of the global coordinate system Rz, kN
Comparison of solutions:
|
Parameter |
Theory |
SCAD |
Deviation, % |
|---|---|---|---|
|
R1, kN |
12.440 |
-10.7735∙cos150° + 6.2201∙sin150° = 12.440 |
0.00 |
|
R2, kN |
0.893 |
0.7735∙cos30° + 0.4466∙sin30° = 0.893 |
0.00 |
|
R3, kN |
5.770 |
5.7735∙sin90° = 5.774 |
0.07 |
|
R4, kN |
0.000 |
0.000 |
0.00 |
|
R5, kN |
0.000 |
0.000 |
0.00 |
Notes: In the analytical solution the reactions in the springs Ri are determined by the quadratic programming method.