Structural Frame Real World Deflection

Question

In the real world, would this pinned frame exhibit a deflection in the fixed, left hand side column?

Arguments from peers largely surround:

1) There wouldn't be left-column horizontal deflection because the moment cannot apply through the pin to the column.

2) There would be left-column horizontal deflection because although the moment doesn't travel through the left-column pin, in reality the right side will sag and pull the left-column towards the right.

Exceptional answers will explain if points 1) & 2) are correct; provides additional information we haven't considered; and the inclusion of visual diagrams if relevant.

Answer 1

Yes the left column would deflect in the real world. The reasons are:

1) Eccentricity of the connection / column:

In reality the load will not be transferred perfectly to the center of the column nor will the column be perfectly straight. This will cause a small amount of bending in the column. In most cases this will be extremely small. However, if the column is slender and the load is large this will cause buckling to occur significantly below the theoretical Euler buckling load. The variation in cross-section is accounted for in the building code material factors. Geometric imperfections must be explicitly checked. For example: Clause 5.2 in Eurocode 2.

2) The connection is not a true, friction-free hinge.

Here is a picture of a typical steel 'pinned' beam-column connection:

Clearly, this is not a friction free hinge and some bending will be transferred to the column. This bending transfer will be small assuming the deflections are small (which they typically are in structures). There is usually some play in the connection as well which will allow some movement. But the bending transfer will not be zero.

2) There is no such thing as a perfectly fixed support.

The foundation has some (non-infinite) stiffness, therefore the fixed support could rotate a small amount. An interesting exercise for the keen reader would be to consider a concentrated moment on an infinite half space made from sold steel (or another material) to get the foundation stiffness then see how this affects the deflection of a cantilever. But I digress...

Ultimately, as with many things in engineering these deflections are very small in typical design situations and are therefore not usually explicitly calculated/considered. However, they do exist. We should not forget the idealisations we make for calculation convenience are simplifications which could have real consequences if not understood.

Answer 2

1) Theoretically this is true, there would not be any horizontal deflection as the fixed supported (left column) would offer complete rigidity and the hinge connection at node 2 would not transfer any bending. There would also be no horizontal force from the top member as the force is completely vertical:

Source: SkyCiv Frame Analysis Software

However, in real life this would not be true.

2) Would be true in reality. This is because the fixed support is not going to offer complete rigidity about node 2 like it does in the above diagram. As the top member deflects, it will cause some forces to transfer in the x direction, not just the y. For applications in structural engineering you would generally consider this negligible.

Answer 3

mg4w's answer has some great examples of how your modelling assumptions aren't true in reality. But even ignoring all of those there will still be deflection in the left hand column.

You may not find any deflection in your analysis, see for example Sam's answer. But this is because you're doing an over-simplistic analysis.

We often assume that we have first-order linear static behaviour. This is because it is easier/faster and normally gives sufficiently useful results. However, a second-order geometrically non-linear (GNL) analysis is closer to reality.

If you hold a piece of string, with a weight applied in the middle, you will find that the ends of the string are being pulled inwards. This is because string has very low bending stiffness, and instead axial stiffness governs. With the top beam in your example it is likely that bending stiffness is significantly greater than axial stiffness, but there will still be some component of axial load. GNL analysis will pick this up. The axial force in the beam will apply a horizontal force on to the top of both columns; this horizontal force will cause the columns to deflect inwards.

Answer 4

Although all of mg4w's points are good, let me add another one: let's say you set up an experiment to determine experimentally whether there was a deflection or not. How would you know if there was a deflection or not? You would have to use some measuring instrument, and that instrument must have a minimum range. E.g. if you used a tape measure, you could detect motion of 1/4", but a deflection of 0.001" might not be noticed. With a dial indicator or LVDT, you could detect 0.001", but maybe not 1e-6 inches. With a laser, you could probably measure 1e-6 inches, but not 1e-12. So experimentally in the real world, the question "Is there a deflection?" is ill-posed, because it cannot be answered definitely. If your instrument reads zero, it could be because there really is a deflection, but it's smaller than your instrument can read. The best you can say is "Is the deflection larger than X?", where X is the smallest resolution of your measurement device.

More specifically though, all real pin joints will transmit some non-zero moment. There is always a non-zero coefficent of friction. It might be a very small amount of moment, but it is not zero. Therefore, yes, there will be some deflection. It is probably so small that you can ignore it for many practical purposes, but it is not zero.