Statically Equivalent Loads

When it comes to numerical integration, OpenSees users either pay too much, or too little, attention. Me? I pay way too much attention to the topic. How else did OpenSees end up with so many integration methods for frame elements? But numerical integration is one of the concepts that users of OpenSees, or any other finite element analysis software, must understand. … Continue reading Statically Equivalent Loads →

Getting the Digits

In a recent post, I pointed out that the LobattoBeamIntegration class uses only ten, instead of 16, significant figures for the locations and weights of the integration points. Those six missing digits made for a perfectly fine demonstration of how to use the isclose function for verification. But, in the bigger picture, omitting six significant … Continue reading Getting the Digits →

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? →

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 →

A Simple Solution to a Complicated Equivalent

A previous post posited on the equivalence of discrete flexural springs (moment-rotation) with integration of continuous moment-curvature response. To find the answer, we can use the principle of virtual forces (PVF) and numerical integration of the internal virtual work: $latex {\displaystyle \int_0^L \kappa(x)m(x)\: dx \approx \sum_{i=1}^N \kappa(x_i) m(x_i) w_i}$ where $latex m(x)$ is the "virtual" … Continue reading A Simple Solution to a Complicated Equivalent →

A Solution, Just Not The Solution

Force-based elements satisfy equilibrium in strong form, even with member loads. However, this does not mean force-based elements always get the exact solution. Consider a simple prismatic, linear-elastic beam with a point load at mid-span. Using a single force-based element with a single point load applied to the element using the eleLoad command. E = … Continue reading A Solution, Just Not The Solution →

A Complicated Equivalent

Whether you use closed-form or numerical integration, the deflection at the free end of a laterally loaded linear-elastic, prismatic column is known to be $latex PL^3/(3EI)$. This result is easily verified in OpenSees with either an elasticBeamColumn element or a material nonlinear element with elastic sections. Now suppose we philosophized a bit then posited the … Continue reading A Complicated Equivalent →

Non-Prismatic Frame Elements

Long before the BeamIntegration abstraction, there was only Gauss-Lobatto integration for force-based elements, with a single section model copied to each integration point. This made it impossible to use a single element to simulate the response of an RC member with different reinforcing details along its length, or any member that was inherently non-prismatic. While … Continue reading Non-Prismatic Frame Elements →

Heavy as a Chebyshev

Numerical integration, or quadrature, is essential for material nonlinear finite element formulations. Gauss, Gauss-Lobatto, or a plastic hinge approach is all you need for frame elements. And for fiber sections, midpoint integration gets the job done. Besides some highly specialized cases, there's no need to use other types of numerical integration in nonlinear structural analysis. … Continue reading Heavy as a Chebyshev →

Nonlinear Elements, Elastic Sections

Using nonlinear elements, particularly the forceBeamColumn element, with elastic sections is just as good as, if not better than, using the elasticBeamColumn element for many reasons. Not only do force-based elements with elastic sections make the transition to material nonlinearity easy, they also facilitate debugging your model. Another reason I like force-based elements is you … Continue reading Nonlinear Elements, Elastic Sections →