Prestressed Members in Flexure

Load stages

A prestressed member is checked at two critical stages:

Permissible stresses (compression and tension) at each stage depend on the member class (IS 1343 Type 1 — no tension; Type 2 — tension without visible cracking; Type 3 — limited crack width).

Stresses in a section

With compression positive, for a beam with prestress $P$ at eccentricity $e$ below the centroid, moment $M$ (sagging), section moduli $Z_t$ and $Z_b$:

Each term: uniform compression from $P/A$; bending from eccentric prestress (hogging, which compresses the bottom); bending from loads (sagging, which compresses the top).

Kern points and the pressure line

Kern (core) distances — the limits within which a compressive force produces no tension anywhere in the section:

For a rectangle of depth $h$: $k_t=k_b=h/6$ (middle third).

Pressure line (line of thrust): the location of the resultant compressive force in the concrete. Under external moment $M$, it shifts upward from the tendon by

• If the pressure line stays within the kern, the section has no tension.
• At transfer (small $M$), a large eccentricity keeps the pressure line near the bottom kern; under service load it moves up towards the top kern — prestressed beams use the whole kern range efficiently.

Cable profiles

• Simply supported beams under UDL: parabolic profile with maximum eccentricity at mid-span and zero (at centroid) at the supports, matching the bending moment diagram.
• Point loads: draped (harped) straight segments.
• Continuous beams: prestress induces secondary reactions and moments. A concordant cable produces no secondary reactions; any profile obtained from the bending moment diagram of the continuous beam under some loading is concordant. Linear transformation — moving the cable at interior supports without changing its shape within spans — does not alter the pressure line (Guyon's theorem).

Magnel's graphical method

For a chosen section, write the four stress limits — top and bottom fibres, at transfer and at service — as inequalities. For example, the top-fibre tension limit $f_{tt}$ at transfer:

Written this way, each of the four conditions is a straight line in the $e$–$(1/P)$ plane. The region satisfying all four gives feasible combinations; the design usually takes the maximum practical eccentricity (limited by cover) and the corresponding minimum prestress. This is Magnel's diagram.