Prestressed question
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user 32699
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The stress from prestressing at the bottom flange of the section is P*e/S + P/A. See figure 7.
First, Pe in the problem is misleading since the "e" subscript is used to mean effective. So the bending effect of prestress is Pe*e/S.
e=h/3
The area to use for this problem is the area of the beam (40in^2). The problem says to assume losses occur before the deck, so the correct area is the girder area (no creep into the deck section)
However, the S to use is the composite section since that is there to resist bending. This is given in Ex 3.18 as 135in^3
Just coincidentally for the girder S=bh^2/6=66.65in^3. In this problem ONLY (don't apply this as a rule) the S to use for the composite section is (nearly) equal to 2 times the S for the girder.
So take the prestress equation above and put all that in in terms of the girder.
P*e/S is equal to P*(h/3)/(2*b*h^2/6) which reduces to P/b*h = P/A
Then there is the P/A from that portion of the equation to get the 2P/A.
So I think the math works, but I totally get the confusion.
As an aside, I don't understand these kind of reductions. What is the point? Prestressing extreme fiber stress is always P*e/S+P/A with proper signs. There is no reason to combine those terms. A similar one I see all the time in geotech reports is pci for subgrade modulus. While this is mathematically correct, psi/i, the intent is the pressure to displace one inch (not to be confused with a density or unit weight).
Good luck!
First, Pe in the problem is misleading since the "e" subscript is used to mean effective. So the bending effect of prestress is Pe*e/S.
e=h/3
The area to use for this problem is the area of the beam (40in^2). The problem says to assume losses occur before the deck, so the correct area is the girder area (no creep into the deck section)
However, the S to use is the composite section since that is there to resist bending. This is given in Ex 3.18 as 135in^3
Just coincidentally for the girder S=bh^2/6=66.65in^3. In this problem ONLY (don't apply this as a rule) the S to use for the composite section is (nearly) equal to 2 times the S for the girder.
So take the prestress equation above and put all that in in terms of the girder.
P*e/S is equal to P*(h/3)/(2*b*h^2/6) which reduces to P/b*h = P/A
Then there is the P/A from that portion of the equation to get the 2P/A.
So I think the math works, but I totally get the confusion.
As an aside, I don't understand these kind of reductions. What is the point? Prestressing extreme fiber stress is always P*e/S+P/A with proper signs. There is no reason to combine those terms. A similar one I see all the time in geotech reports is pci for subgrade modulus. While this is mathematically correct, psi/i, the intent is the pressure to displace one inch (not to be confused with a density or unit weight).
Good luck!
user 32699
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Correction to the above.
The correct S to use for the prestress in the bottom of the section is the S of the girder since the section is not composite yet.
In the composite section, without creep, the correct S for prestress is that of the girder, the composite modulus is used to resist loads.
The prestress is applied at h/3, the eccentricity is e=h/6.
So, P*e/S is equal to P*(h/6)/(b*h^2/6) which reduces to P/b*h = P/A
The correct S to use for the prestress in the bottom of the section is the S of the girder since the section is not composite yet.
In the composite section, without creep, the correct S for prestress is that of the girder, the composite modulus is used to resist loads.
The prestress is applied at h/3, the eccentricity is e=h/6.
So, P*e/S is equal to P*(h/6)/(b*h^2/6) which reduces to P/b*h = P/A
David Connor SE
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I agree with VTBridge as well. Combining P/A + M/S terms in these prestressing problems just makes things more confusing and you forget if you are dealing with compression, tension, top, bottom, etc. Also, watch out for sign conventions. For prestressed, compression is (+) and tension is (-). I've seen some examples switch that around or refer to a "minimum" tension which is actually the higher tension stress because it's negative. Keep stresses separate until you combine them in the last step with compression (+) and tension (-) would be my advice.
