Beam Displacement - End translation

[slickdeals]

Folks,
I am trying to determine how much longitudinal translation occurs in a beam when it deflects vertically. I am doing this to see if displacement under wet weight of concrete will cause the beam to move more than the tolerance of a standard hole beam.

I tried to do a quick run in ETABS/SAP, but they would give me only end rotation and not translation.

 

[slickdeals]

I tried again by introducing an in-plane kink (imperfection), but I am not sure what magnitude of imperfection to put in beams to account for a lateral translation at ends.

 

[paddingtongreen]

Theoretically, the central fibers remain the same length, but they describe an irregular arc in the deflected beam. Fake it by putting a circular arc through the connections and the central deflected point. Calculate the new separation of the supports. Even though this is a conservative estimate, I forecast that it will be insignificant.

On the other hand, if your calculus is up to it, you could an exact analysis

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[Lion06]

Are you concerned about movement at the bottom of the section, centroid, or top? The answer will be different for each. You could do a really rough check using a triangle with half the length as one side, the midspan deflection as the other, then see
what difference is between the hypotenuse and half the length. That's for the midspan horizontal movement since it doesn't give he movement do to end rotation. Can you build the beam out of plate elements and load it? That should give you what you want from the software.

 

[Lion06]

This is obviously a lower bound because the deflection is not a straight line as indicated in the procedure above, but it should give a
reasonable approximation with relative ease. I just a quick check for a 20' beam. Assuming 3/4" max deflection due to wet concrete, I'm coming up with a lateral translation of 0.0023" - essentially 1/400"!!!

 

[slickdeals]

@SEIT:
I did a shell element model of a W24x55 beam (with proper web and flange thicknesses) spanning 34'. The wet weight of concrete was 600 plf. The shell element model was supported along the centroid using pin and roller.

The mid-span deflection of a W24x55 frame element with 6" mid span kink produced a vertical deflection of 0.48" and a lateral translation of 0.0287".

A shell element model produced a vertical displacement of 0.42" and a lateral translation of 0.0395" (top), 0.000615" (centroid) and 0.038255" (bottom).

 

[rb1957]

if you've got a simple cantilever and a simple load (UDL?), you can easily find M(x) and integrate twice to get EI*d(x). then assume the deflected shape is a circular arc (i'm not clever enough for anything else) then R*theta = L and cos(theta) = (R-d)/R ... i wonder if theta is related to the slope at the end of the beam, v(L) ? then L-R*sin(theta) is the change in length of the NA, and to account for the rotation of the end face, the change in length of the lower chord is dL+yNA*theta (or 90-v(L), maybe ...)

 

[Splitrings]

I first thought was the change in length of the outer fiber would be theta*half the depth of the beam. Which by my calculations is roughly 0.21". Then I opened my structures text and it states "...moment causes the bottom fiber of the beam to be elongated by delta.". So that would be roughly 0.45". Seems like a lot. Slickdeals, is it possible your elongation values are in feet. If so they would be roughly 0.47".

 

[slickdeals]

I double checked my model. The value I reported are in "inches" and not feet.

 

[slickdeals]

And I am not sure if you can directly equate how much the bottom fiber elongates vs. how much it translates at the support.

A beam with pinned ends (no lateral translation) will still elongate, right?

 

[Lion06]

Slick-
I agree. The movement of the top and bottom of the section are due a lot more to end rotation of the beam than to actual horizontal translation. The only place that undergoes pure translation is the neutral axis, which was extremely small.

 

[ToadJones]

for the record, this is purely academic, no?

 

[slickdeals]

Toad:
Unfortunately it is not. It is the real deal. I cannot, however, go into specifics.

 

[ToadJones]

I imagine a general format for the answer could be set up for any beam given the depth, length, and deflection at the center, given symmetry as well.

 

[paddingtongreen]

Since this is a real deal, is there control over the wet concrete thickness at the center, or is the top leveled despite the deflection of the beam?

You will need upper and lower bound answers for beams with high and low side rolling tolerances.

Then again, the bolt might be hard on the edge of the hole to start with.

Sorry to sound so negative, but I get worried when we speak of such small dimensions. My own view is that steel is very forgiving and will try to do the job even if we don't quite get it right.

If you divide the beam into say six or eight parts ans calculate the deflections at each, and then the reduction in horizontal distance between the points you could calculate a new horizontal length between supports.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[Elabbasi]

There are two sources of longitudinal deflection in a vertically loaded horizontal beam. The first is the deflection of the neutral axis which should be zero if linear beam theory is used, and non-zero otherwise to keep the beam length approximately constant. The second is the deflection relative to the neutral axis which is equal to slope x distance from the axis.

Depending on the geometry and loads one component of longitudinal deflection could be larger. My guess is that for the beam in question the second component is more significant as StructuralEIT mentioned.

If you are using a computational software such as ETABS/SAP the first component can only be obtained if you perform a nonlinear analysis. The second component you have to calculate using the slope that the program calculates (or from beam theory).

Nagi Elabbasi
Veryst Engineering

 

[rb1957]

i happened to notice there's something in Roark 7th Ed on this topic, under (predictably enough) the chapter on beams.