huangyhg

a seemingly simple question but in fact very difficult?
please see attached file.
the base plate is adequate. the 300 kips-in overturn moment can be in any direction.
required: what is the maximum tension (kips) in anchor bolt.
note: only left hand drawing is in original question. the right hand drawing is based on my analysis, it may not be correct. you can ignor it.

check at 0 degrees (2 bolts and concrete triangular dist)
& 45 degrees (1 bolt and concrete triangular dist)
talk about an easy question turning hard, i just love the semi-rigid designs. before i attempt an answer to your question, i will need to know a few things.
1. what is the requirement for stiffness? i.e. do you have a maximum assumed rotation?
2. do you have an uplift force?
3. what grade bolt are you using?
4. base plate thickness?
you may ask why do i want to know all this additional information, and the reason is in semi-rigid design there are a few methods for design, i personally like for the base plate design the component method by wald f et al. and these inputs allow you to select you plate mode. ie prying ect.

when in doubt, just take the next small step.

i'm getting around 27k of tension for the case where it is at 45 degrees to the baseplate. i also needed to use 8ksi concrete for that, anything less and it won't even work.
assuming what? plane sections remain plane?
marked for later review.
similar problems come up with anchored tanks and stacks.
one problem is that people want to over-analyze it. the appoach used in the tank codes is to assume that stress distribution in the tank/stack is mc/i, and then assume that anchor loading must be distributed similarly, so the anchors are just treated as a ring of metal of equivalent area. and this seems to be carried to its logical conclusion in the large transmission towers that are supported by the bolts, with no contact between the base plate and concrete.
the alternate approach is to take a section immediately below the surface of the concrete, and analyze it as a composite steel-concrete section, and derive the anchor distribution from that assumption. the flaw is that anchor loading immediately above the concrete must be the same as immediately below the concrete, so making assumptions that give you different numbers above and below doesn't really gain anything other than complication.
in the case of your right-hand sketch, i think the flaw is that you have no rational basis to determine that triangular loading. in fact, the base plate is flexible and 2-dimensional, and contact stresses would very across the face, not just linearly along an axis. there is also going to be some amount of bolt pretension, and that unknown amount of pretension will also result in a distribution of contact stresses across the bolt area. so you can make one assumption and calculate bolt loads simply. you can make a different assumption and work through a bunch of math and come up with a bolt load, but it's still based on your assumptions. it's not immediately clear if you've gained anything by the complication in that case.
two solutions come to mind. one is to refer to standards governing the type of construction, and see if there is a codified approach. this won't necessarily be more accurate than whatever solution you derive on your own, but using a method and an allowable stress that have proven to be satisfactory is an acceptable approach, even if the theory leaves something to be desired. the second solution is to support the base plate on the bolts, which simplifies the problem. there are many cases where it is easier to change the geometry to fit the math than the other way around.
i'd assume the compression field is tangent to the cyclinder. the centroid position of a triangular dist'n on a triangular base would be nice to work out (even though someone else (roark?) has already done it.
but would you allow the baseplate to crease along edge of the compression field ? are you looking for the ultimate strength (plasticity) or a near enough approximation, in which case plane sections remain plane would be reasonable.
how do you take bolt preload into account ... on the tension side it's reasonably easy (gapping, physical extension of the bolts, rotation of the baseplate) ... on the compression side ?
have you tried limit states design (lrfd) approach. problem becomes much simpler and with the concrete in confined compression.
jstephen-
it is common practice to assume the base plate is rigid. additionally, you can only count on an equivalent area of steel for the anchor rods if there is some mechanchism by which the rods can take compression. this may be common for transmission towers/lightpoles with nuts under the plate, but is not common for the large % of baseplates.
i used the lrfd approach and assumed a rectangular pressure distribution (not triangular) with the magnitude = phi*(0.85)*f'c. the compression area was so large (8.5" - even with 8ksi concrete) that only one anchor was in tension. technically, it would have had (3) in tension, but two are so close to the na that it is reasonable to neglect them.
if you take a cross section through the center of that base plate, and look at the moments about the left side of it- you have the anchor force times the moment arm. you have half of the bending moment in the column (assuming the mc/i force distribution in the column). and you possibly have a bending moment in the base plate itself. note that the diagram doesn't show a vertical load, so i'm assuming that the left side of the base plate is not experiencing compression forces from the concrete.
if you assume that the bending moment in the base plate is negligible, then you can immediately calculate the anchor force, which will be the same as if the column were supported on the bolts. if you assume that the bending moment in the base plate is significant, then the assumption of a rigid base plate would appear to be questionable. or you could assume that bending in the base plate is negligible, and assume that bending stresses in the column do not follow the mc/i distribution, but then you're left without a column analysis. so it seems to me you can work this different ways, but it is not obvious which is actually better- all involve rather gross approximations.
note also that for the example shown, with pure moment, if the compressive loading turns out to have its centroid out beyond the far bolts, then you get a less conservative approach than assuming the load on the bolts only.

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