Semi-rigid connection

Results.

Problem. A semi-rigid connection is a rotational spring between the element end and the node. A zero spring stiffness gives a release and an infinite one gives a rigid connection.

Reference document. The series spring relation is used, with the force method for the fixed-end forces. The case with two springs is cross-checked with an independent condensation.

In a semi-rigid connection the end moment is proportional to the relative rotation between the element end and the node.

The connection spring and the end stiffness of the element combine like two springs in series.

Numerical model. The geometric stiffness and the mass matrix pass through the same link, so buckling and modal results are tested as well. In the twin-node model the translations, $r_x$ and $r_z$ are tied and only the in-plane rotation $r_y$ is left independent.

End rotational stiffness, springs in series#

Parameters. 4EI/L = 10,666.7 · L = 6

Closed form. Springs in series

Note. The result is exact at all four spring stiffnesses between $10^2$ and $10^8$.

Uniformly distributed load, spring stiffness 10,000 at end i, end j rigid#

Parameters. w = 12 · L = 6

Independent calculation. The force method, with the spring entering the flexibility diagonal as $1/S$

Note. At the rigid end the moment is $wL^2/12 = 36.00$.

Uniformly distributed load, spring stiffness 10,000 at both ends#

Parameters. w = 12 · L = 6

Independent calculation. The force method, with the spring entering the flexibility diagonal as $1/S$

Note. At the rigid end the moment is $wL^2/12 = 36.00$.

Uniformly distributed load, spring stiffness 5,000 and 20,000 at the ends#

Parameters. w = 12 · L = 6

Independent calculation. The force method, with the spring entering the flexibility diagonal as $1/S$

Note. At the rigid end the moment is $wL^2/12 = 36.00$.

One node, two different ends#

Parameters. S = 8,000

Closed form. The definition itself, with the end rotation and the end moment obtained along two separate paths

Note. The nodal rotation is -3.683036e-03 while the end rotation of the sprung member is -1.004464e-03. The rigid neighbour at the same node shares the nodal rotation exactly.

Portal frame, very stiff spring limit#

Parameters. S = 1,000,000,000,000

Independent calculation. The rigidly connected frame obtained as the spring stiffness goes to infinity

Note. A rigid connection gives 1.11933e-02 and a hinge 3.55681e-02, and the intermediate spring stiffnesses ($10^3$, $10^4$, $10^5$) give 2.52880e-02, 1.41323e-02 and 1.15230e-02.

Pin-ended column, buckling#

Parameters. mesh = 16 elements · L = 4

Published source. The critical load of a cantilever column

Note. The convergence runs from 937.50 down to 925.28, from above and monotonically.

Beam with a hinge at mid-span, modal analysis#

Parameters. mesh = 8 elements · L = 6

Independent calculation. An independent build, with twin nodes and an equality constraint on the translations

Note. The first three vertical modes are 71.9495, 112.4513 and 288.9744 Hz with condensation and 71.949, 112.4344 and 288.8568 Hz with twin nodes. The difference comes from Guyan reduction.