Prestressed Beam Bottom-Fibre Concrete Stress
Prestressed Beam Bottom-Fibre Concrete Stress
Prestressed concrete beams work by intentionally pre-compressing the concrete with tensioned steel strands before service loads are ever applied, so that when bending moment tries to put the bottom fibre into tension, it first has to overcome the compression already stored there. The bottom-fibre stress calculation checks whether the strand force, its eccentricity below the beam centroid, and the applied moment combine to keep that fibre in compression (or within an allowable tension limit).Getting the fibre stress calculation wrong — sign errors on the moment term are a classic mistake — can miss that the bottom fibre is actually cracking under service load, since prestressed sections are typically detailed to be uncracked in service; once cracking occurs, durability (corrosion protection of the strands) and long-term deflection behavior both degrade compared to the design intent.
The bottom-fibre stress combines three superimposed effects: direct axial compression from prestress, added compression from the prestress eccentricity, and tension relief (or addition) from the applied moment: sigma_bot = P/A_c + P·e/S - M/S. where P is the prestress force, A_c is the concrete cross-sectional area, e is the eccentricity of the prestress below the section centroid, S is the section modulus at the bottom fibre, M is the applied bending moment, and sigma_bot is the resulting bottom-fibre stress (positive taken as compression).
Inputs
P := 1000 kN=1e+06 N
A_c := 0.15 m^2=0.15 m^2
e := 0.15 m=150 mm
S := 0.01125 m^3=0.01 m^3
M := 250 kN·m=250000 Nm
The prestress force acting alone would compress the whole section uniformly, and its eccentricity below the centroid adds extra compression at the bottom fibre through a bending-like term; the applied moment then works against that stored compression by pulling the bottom fibre toward tension.
sigma_bot := P/A_c + P·e/S - M/S=-2.22 MPa
Results
A negative result, as found here (about -2.2 MPa using the compression-positive convention), means the bottom fibre is actually in net tension under this load combination — the applied moment has overcome the stored prestress compression at this fibre. If the concrete's cracking tensile strength is exceeded, the section may crack under this load case even though it was designed to remain largely uncracked, which is a red flag for durability and deflection performance that would normally prompt increasing the prestress force or eccentricity. This single check should be run for every governing load combination (including the smaller moments seen right after transfer, before losses), since the fibre that governs can change between transfer and full service load.
Assumptions & limitations
- Uncracked, linear-elastic concrete section behavior assumed (transformed section effects of the strands are neglected).
- Prestress force P is the effective (post-loss) value; short- and long-term losses are assumed already applied.
- Sign convention: positive stress is compression, negative is tension.
- Only the bottom fibre is checked; the top fibre requires the mirror-image calculation with its own section modulus.