This is a very easy calc. Do a search on here, it has been covered many times already.
A quick synopsis is as follows.
Find the PNA by locating the axis about which there is an equal area of steel above and below. Basically, divide the total As by 2 and determine what axis will satisfy that condition - That is your PNA.
Next find the centroid of the area above the PNA and the centroid of the area below the PNA.
Now multiply the area of steel above (or below - the will be equal) the PNA by teh distance between the centroids of teh areas above and below.
This will give you the plastic sections modulus in in^3.
It should be noted this is only valid for a homeogenous material such as all steel with the same yield stress. If you have a different yield stress (which it sounds like you probably do with the plate below - you can spec a 50 ksi plate ) or follow the same procedure as above with a slight modfication.
Instead of locating the PNA where there is an equal area of steel above and below, locate it such that you satisfy statics above and below (i.e. find the axis about which when the entire section yields you have an equal C and T). This shouldn't be that hard to do and I don't feel like writing out an equation. I would find Mp first in this case and back calc into Z.
Sum moments of each portion of the cross-section about the PNA (making sure to use the correct Fy for the plate if it happens to be 36 ksi) and that will give you Mp. I'm not sure if you can get a true Z since Fy is not constant.
Be certain to account for the different Fy of the plate unless you spec a 50 ksi plate which is uncommon. If you follow the typical procedure for calc'ing Z, you will be counting on 50 ksi from the plate when it can only give you 36 ksi.
The section I am trying to modify is an existing VERY old beam with an Fy of only about 28ksi. I'm adding on a 36ksi plate, so I'm just being conservative and saying that the plate is also 28ksi unless I have big issues and cannot get that to work.
got it!
if you can't get the plate to work, an option I have used in the past to reinforce exisiting steels beams is to use a WT for reinforcement instead of a plate. It works quite well.
If you're trying to be conservative, I would say to use the elastic section modulus Sx instead of the plastic modulus Zx. With a wide flange, you're only gaining 13% or so going from Sx to Zx. A little conservatism is a good thing, esecially if you're messing with old members where there may be other things to consider that you might not think of right away.
StructuralEIT, why do you say 50 ksi plate is uncommon? Many of the fabricators we work with use it all the time, often replacing A36 plate for Grade 50 in the shop.
and dont forget that reinforcing will only serve to take live loads unless the member is unloaded (with shoring or by other means) during installation.
The original beam will maintain the stress from dead loads and the reinforcement + original will act as one to take the loads that were not present during installation.
