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 →

Plate Rebar Material

Where fiber sections integrate stresses over two dimensions for beam-column line elements, fiber sections integrate stresses through only one dimension for shell elements. Either way, you're performing volume integrals, whether it's two dimensions in the section and one in the element or one dimension in the section and two in the element. The LayeredShellFiberSection, where … Continue reading Plate Rebar Material →

Is It Close Enough?

The locations and weights for Gauss-Lobatto beam integration, the de facto beam integration for force-based elements, are hard-coded in the OpenSees source code. For most cases in the LobattoBeamIntegration class, the locations and weights are written to only ten significant figures instead of 16 or more. Although I am certain that leaving six sig-figs on … Continue reading Is It Close Enough? →

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 →

Murum, cura te ipsum

OpenSees has its fair share of element implementations that are computationally inefficient. Fortunately, most of those elements are never used. But among elements that are used, SFI-MVLEM is the undisputed champion. Whereas the standard MVLEM element uses a uniaxial material in each fiber, the SFI-MVLEM element accounts for the interaction of axial and shear stress ($latex \sigma_{11}$ … Continue reading Murum, cura te ipsum →

Minimal MVLEM Example

Due to its simplicity and efficiency in modeling shear walls, the MVLEM (Multiple Vertical Line Element Model) is among the more frequently asked about elements in OpenSees. The MVLEM is also one of the better documented elements in OpenSees with documentation for both its 2D and 3D versions. Personally, I haven't used the MVLEM for … Continue reading Minimal MVLEM Example →

Beam With Hinges Interior Redux

When it was but a wee blog, I wrote a post expressing second thoughts on how I presented the elastic interior of the beamWithHinges element in Scott and Fenves (2006). The post had a lot of words, but no figures or examples. So, let's try again. Consider a simple span comprised of one beamWithHinges element, … Continue reading Beam With Hinges Interior Redux →

It Takes Two

While it might be advantageous for some modes of response in some solid finite element formulations, using a single integration point in a frame element is not OK because the element needs to integrate quadratic polynomials--something you just can't do with one integration point. The result of using one integration point is generally pretty bad … Continue reading It Takes Two →

Could Not Invert Element Flexibility

Due to inversion of section stiffness to flexibility, then inversion of element flexibility to stiffness, the force-based frame element implementation in OpenSees is finicky, to say the least. A common error message spawned from the force-based element is "could not invert flexibility..." You will see the same message many times, but, ultimately, you will get … Continue reading Could Not Invert Element Flexibility →