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Bottom Fiber Stress in a Prestressed Beam

Mediumstresseccentricitysuperposition
An uncracked gross section has area A = 0.30 m² and bottom section modulus Sb = 0.05 m³. Effective prestress Pe = 900 kN acts 0.15 m below the centroid. An external sagging moment M = 150 kN·m acts at the same section. Using compression positive and linear superposition, what is the bottom-fiber stress?
  1. A8.70 MPa compression
  2. B2.70 MPa compression
  3. C2.70 MPa tension
  4. D5.70 MPa compression
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Answer: B. 2.70 MPa compression

σb = Pe/A + Pe·e/Sb − M/Sb. Pe/A = 900/0.30 = 3,000 kPa = 3.00 MPa. Pe·e = 900(0.15) = 135 kN·m, so Pe·e/Sb = 135/0.05 = 2,700 kPa = 2.70 MPa (compression at the bottom). Sagging M causes bottom tension: 150/0.05 = 3,000 kPa = 3.00 MPa. σb = 3.00 + 2.70 − 3.00 = +2.70 MPa compression. 8.70 MPa adds the sagging term as compression (sign error). 2.70 MPa tension flips the prestress-eccentricity sign (3.00 − 2.70 − 3.00 = −2.70). 5.70 MPa ignores the external sagging moment.
MPa (compression)
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