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

# Tag: Numerical integration

# 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

# Force-Based Beam-Column Integration Options

I like numerical integration because it allows you to do a lot of interesting things with force-based frame elements--so much more than simulating the response of reinforced concrete moment frames. Numerous numerical integration options are available in OpenSees, so in 2011 I wrote and uploaded to the OpenSees wiki a PDF summarizing those options. According … Continue reading Force-Based Beam-Column Integration Options

# Integration Points with Negative Weight

A colleague in Eastchester once told me that faculty have three, maybe four, good ideas over their career. In other words, a faculty member could have over a hundred papers, but there's only three or four underlying concepts. Perhaps it was "two, maybe three", but you get the point. Playing with integration points and weights, … Continue reading Integration Points with Negative Weight

# More Is Not Always Better

I sometimes run across simulations where frame member response is computed using displacement-based beam-column elements with more than two Gauss points per element. These elements require at least two Gauss points to ensure a complete solution and to capture the exact solution for a linear-elastic, prismatic member. While it is well known that you can … Continue reading More Is Not Always Better