Buckling of Restrained Plates

The plate buckling analysis in a previous post demonstrated flexural buckling–definitely minimal and definitely necessary if you’re going to tackle more complex cases of plate buckling. Having passed the minimal example, Mark Denavit, a frequent collaborator, suggested I take a look at buckling of plates that are restrained on all four sides. Mark always has good … Continue reading Buckling of Restrained Plates →

Minimal Plate Buckling Example

OpenSees is not built to perform linear buckling analysis. But a few years ago, Luigi Caglio shared a workaround described in this post. In the post, the example application is a frame model, but there’s no reason the approach cannot work for shell models. So, here’s a minimal working example. Consider a rectangular steel plate with simple boundary … Continue reading Minimal Plate Buckling Example →

Critical Buckling Loads via CBDI

Curvature-based displacement interpolation (CBDI) is a method of approximating the transverse deflection at each integration point, or section, of a force-based frame element in order to account for geometric nonlinearity within the basic system, i.e., "P-little-delta" effects. Neuenhofer and Filippou (1998) describe the complete CBDI formulation for force-based elements. But one piece of the formulation, … Continue reading Critical Buckling Loads via CBDI →

Corotational Rigid Offsets

Unlike Linear and PDelta, rigid joint offsets are not an option for the Corotational geometric transformation in OpenSees. And the lack of corotational rigid offsets is not due to theoretical limitations, simply no one has taken the time to implement the equations. The two-dimensional case would not be terrible. But three-dimensional offsets? No thanks, not … Continue reading Corotational Rigid Offsets →

Torsional Buckling

From time to time, I dabble with the OpenSees warping elements developed at the University of Sydney in 2011 for doubly symmetric sections, then in 2016 for open sections. Since my last foray into warping, I've taken the nonlinear displacement-based formulation from "it compiles" to "it works". Some other ancillary changes to the warping fiber … Continue reading Torsional Buckling →

One and Only One

Two element formulations in OpenSees--forceBeamColumnCBDI and mixedBeamColumn--are capable of handling geometric nonlinearity within the basic system, i.e., P-little-delta effects. The CBDI formulation, described in Neuenhofer and Filippou (1998), approximates the transverse deflection using Lagrange polynomials fit through the curvature at each integration point. Due to the added computational expense and coding details, the forceBeamColumnCBDI element … Continue reading One and Only One →

Two Paths You Can Go By

I am confident we can use OpenSees to solve every truss, beam, and frame problem from any statics or structural analysis textbook as well as every single degree-of-freedom and rigid shear frame problem from a structural dynamics textbook. We can also solve any reasonable problem from a finite element textbook. My confidence starts to wane … Continue reading Two Paths You Can Go By →

Slender Things

Using fiber sections and the corotational geometric transformation is an easy way to simulate combined material and geometric nonlinearity in column members. A previous post examined this approach for steel columns where residual stresses play an important role in the axial load capacity. In this post, I will show the corotational mesh approach for non-sway … Continue reading Slender Things →

Meshing for Column Loads

For material nonlinear analysis of frame models, you can improve the computed response by using more displacement-based elements or more integration points in a force-based element. The material nonlinearity occurs inside the basic system, also known as the natural system or the kernel. To capture geometric nonlinearity due to large displacements, you have to go … Continue reading Meshing for Column Loads →