Various Questions with Straight-Line Approximation of Curved Beams

Looking for some help regarding understanding the 'right' way to do a straight-line approximation of a curved member, and limitations that need to be accounted for. My questions are near the end and highlighted if you wish to skip it, but understanding the whole picture you might want to read everything.

A colleague of mine has modelled a 'curved' parabolic beam using nodes and straight-line members to approximate the curved shape. They created the curve in the XZ plane (horizontal plane) and loaded it radially in the horizontal plane as well using UDLs in the member's local 'Z' direction. We would have expected to get an accurate representation of the shear force diagram when using tight node spacing, however there were always these large jumps in the SFD:

My interpretation is due to the change in angle between the adjacent straight-line elements is causing these effects, but I cannot count for this magnitude of effect given we used very tight spacing (~70mm over ~8.67m). As noted in other topics regarding 'curved' members - STAAD has a limitation of not being able to calculate intermediate shear/bending moment diagrams for curved members, but always recommended of modelling with straight-line members for approximation. As we have done straight-line members for approximating the curve, I would expect the model is not subject to these same limitations but looks like we are.

I did an experiment and took to heart the limitation as being the case for any curve beam (either approximated by straight lines between nodes, or exact using STAAD's curve member specification). If STAAD (or any analysis program for that matter) cannot calculate intermediate axial/shear/bending for curved members, then the limitation would also be to the load application. So I converted this UDL application to Member Concentrated Load command with 'd1' & 'd2' = 0 to apply the concentrated loads at the starting node of each member to more easily deal with applying the loads radially. This gave a much cleaner SFD:

These diagrams are very close to the SFD and BMD for the approximation of applying loads directly to the nodes, which confirms the jumps in the above SFD can be ignored:

An additional issue we saw was that when changing the section size the shear/bending moments were magnified (the thicker the concrete section, the more shear/bending moment and less axial force). Note that there are no self-weight loads active and only fixed loads are applied, so changing the section size should have no impact on the SFD or BMD. For instance, a PRIS 1000x500 section has a max shear of ~15.4 kN, but a PRIS 1000x1000 section has a max shear of 57.4kN. 

My questions for this post are:

  1. Why does this issue arise when applying radial UDLs to a straight-line approximation of a curved member? We are using straight line members as recommended by other topics / help pages, and I would expect a more cleaner SFD regardless of using a UDL load. Depending on if we take the starting node or ending node side of the straight-line members for the true shear force will give drastically different results. There is also overlap in the SFD, so where Shear = 0 cannot be readily determined.

  2. Do the limitations of curved members in STAAD also apply to straight-line approximations of these curved members? My experiment seems to suggest this, and straight-line approximations of curved members should not have intermediate loads applied.

  3. Why does increasing section size cause magnification of the SFD and BMD? If everything stays the same (geometry, material, loading), then changing the section size and its stiffness should not change axial/shear/bending magnitude.

I have attached a STAAD file that has 6 versions of the curved beam in the same file to see the results simultaneously, using the key diagram below:

 2D Concrete Arch Beam_forum.STD

Any assistance would be great! Looking forward to the discussion. I will note that this is not a STAAD-specific issue, but more an analysis/understanding issue; I copied the same model into Space Gass and the same results come up.

  • So I experimented a bit more, and indeterminacy was the culprit. Releasing one of the supports to allow to roll in the X direction settled down all types to the same axial/shear/bending diagrams.

    For (Q1) - issue is due to the analysis being indeterminate. As long as the structure is determinate, then the type of loading does not matter.

    For (Q2) - seems to suggest that for indeterminate analysis it is recommended to load only at the joints/nodes to avoid odd results in the SFD/BMD.

    For (Q3) - going from a 1000x500 to a 1000x1000 concrete section increases bending stiffness by 8x, but axial stiffness is only increased 2x, so the curved member resists the loading more in bending than axial when pinned at both ends (indeterminate). This is why SFD/BMD increased for the larger section with a reduced compressive thrust. Releasing one support to 'roll' removes the section stiffness dependency when generating the internal member forces.

    Anyway, if anyone has other thoughts on it, feel free to add. Just wanted to post what came out from me looking at this.

  • I haven't had a chance to look into your model. But here is something that may help in understanding the jump in the shear force diagram.

    When you apply a concentrated force at the start or end of a member as a member load using the "CON direction-term" method, that load is treated as acting not at the geometrical node location of that member, but at an infinitesimal distance away from it. So, whether the load is applied at the start or at the end, its point of action is a tiny bit inside the member's span.

    On the other hand, the member end forces that the program reports are at the geometrical start and end of the member. So, when it plots the BM or SF diagrams, it calculates the force at 11 intermediate locations by travelling from start to end. As it travels from the start node location to the next interior point, the shear force experiences a sudden change due to the presence of the concentrated load which is inside the member, albeit only a tiny distance away.

    If the load is applied using the joint load method, you won't experience this jump, because a load applied at the end is at an infinitesimal distance "outside" the member. This is something that is inherent in the stiffness method procedure, which is probably why you see this behavior in SpaceGass too.

    I will have a look at your model when time permits and let you know if I find more clues to the behavior.

    Regarding your question on change in SF or BM due to a change in member size, that is most likely due to shear deformation. STAAD computes shear deformation by default and this affects the node displacements which in turn affect the member end force calculation. The greater the member size, the larger the shear deformation. You could try switching off the shear deformation computation by adding the command "SET SHEAR" on the second line of the STAAD input file, and see if the 2 models still produce different SF and BM values.