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 improve simulated frame response by increasing the number of integration points in a force-based beam-column element, does the same hold true when using more than two integration points in a displacement-based element? Let’s find out using a model of a W14x90 steel member that appeared in a previous post comparing the displacement-based and force-based formulations.

The member resists a constant axial load, N=0.2F_yA, and a moment, M, that increases from zero to two times the yield moment, M_y=F_yS. The section area, A, and strong axis section modulus, S, for a W14x90 are found in the Steel Manual.

Interaction of the axial force and bending moment along the member is captured by fiber-discretized cross-sections. The stress-strain response of each fiber is bilinear with F_y=50 ksi, E=29,000 ksi, and \alpha=0.005.

We will examine the global and local response obtained from one displacement-based beam-column element (dispBeamColumn in OpenSees) with N_p=3, 4, and 5 Gauss-Legendre integration points. The analysis does not include geometric nonlinearity within the element.

The moment-rotation and moment-axial displacement response are shown below. Increasing the number of integration points does not improve the computed solution. Although there are some differences in the post-yield response, neither for better nor worse, the simulation consistently over-estimates the yield point.

The local flexural response at the final load step is shown below. As expected for the displacement-based formulation, the curvature distribution remains linear across the element and there is no significant change in the distribution as the number of integration points increases. The bending moments computed at each integration point do not converge to the exact solution–they merely scurry around to satisfy weak equilibrium.

Similar trends are observed for the local axial response. The axial deformation is constant along the element, as expected, but shifts a bit as the number of integration points increases. Thanks to weak equilibrium, the axial force at each integration point is pretty far from the expected solution; however, the weighted sum over all points is equal to the expected result of 265 kip compression in all cases.

So, my assessment is that increasing the number of integration points in a displacement-based element isn’t going to help you. It won’t really hurt you either; however, the amount of computational time spent on section state determinations can start to add up, especially for fiber sections, and cost you valuable wall time for no added benefit.

Two integration points per displacement-based element is all you need–you’re going to have to refine the mesh anyway.

13 thoughts on “More Is Not Always Better

  1. Hello PD,
    Does the strain hardening ratio of steel material affect on the displacement-based element response? If yes, How we should regularize reinforcing steel response?
    I have tried to determine post-yield energy for regularizing reinforcing steel response using an approach often employed for continuum analysis of unconfined concrete responding in compression. But the results are very bad. The strain-hardening ratio is large for an element with, for example, 60 cm length.
    I think for reinforcing steel in the displacement-based element we should consider length associated with an integration point, but for concrete material, we should use the length of the entire element.
    I will be so grateful if you guide me.
    Best regards,


    1. Hello Mohsen,
      The strain-hardening ratio definitely affects the element response. You’ll have to search the literature for regularization approaches for steel materials. The approaches will be different for concrete and steel.


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