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Question

A prestressed concrete simply supported beam of span 6 m has a rectangular cross-section 300 x 600 mm and unit weight 24 kN/m3. The resultant prestressing tendon has zero eccentricity at ends and an eccentricity, e, at midspan, and linear profile. Under initial condition, the allowable stresses are 20 N/mm2 in compression and 2 N/mm2 in tension. Using this condition, determine the prestressing force, P, and the eccentricity, e, at midspan. The imposed loading is a single concentrated load, W, at midspan and the tendon undergoes loss in prestress of 25%. Determine W, if the final allowable stresses in compression and tension are not to exceed 15 N/mm2 and 1 N/mm2 respectively.

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Solution



Z=bd26=300×60026=1.8×107 mm3

D.L =A×γc=24×0.3×0.6=4.32 kNm

Md=wd×l28=4.32×628=19.44 kNm
In initial condition tension will be at top and compression will be at bottom
ftop=PAPeZ+MdZ
2=P1.8×105Pe1.8×107+19.44×1061.8×107 ...(i)
fbottom=PA+PeZMdZ
20=P1.8×105+Pe1.8×10719.44×1061.8×107 ...(ii)
from (i) and (ii)
P=1.62×106 N
Substituting in (ii)
9 + 0.09 e - 1.08 = 20
e = 134.22 mm
Loss in prestress = 25%
P=0.75×1.62×106
=1215×103 N
In final condition compression will be at top and tension will be at bottom.
ftop=PAPeZ+MdZ+MLZ
1.5=1215×1031.8×1051215×103×134.221.8×107+1.08
+w×6×1064×1.8×107
w=194.75kN
1=1215×1031.8×105+1215×103×134.221.8×1071.08w×6×1064×1.8×107
fbottom=PA+PeZMdZMdZ+MLZ
W = 188.75 kN
The concentrated load at mid span = 194.75 kN.

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