Bridge & Structural Engineering, Post-Tensioned & Pre-Tensioned Concrete
Prestressed Concrete Tendon Kern Point & Fiber Stress Formula
Calculate Section Kern Radius k = S / A (m), Top-Fiber Concrete Service Stress f_top (MPa, Compression +), Bottom-Fiber Service Stress f_bot (MPa), and Class U Uncracked Status.
Placing a tendon below the lower kern k = S / A induces top tension at transfer—until dead+live gravity moment balances it!
In pre-tensioned bridge girders and post-tensioned parking beams of cross-sectional area A (m²) and symmetric elastic section modulus S (m³), clamping Effective Prestress Force P_e (kN) at eccentricity e (m) below the centroidal axis superposes THREE elastic stress terms (compression positive +, tension negative -): Top Extreme Fiber Stress is f_top = (P_e / A - (P_e * e) / S + M_service / S) / 1,000 MPa, while Bottom Extreme Fiber Stress is f_bot = (P_e / A + (P_e * e) / S - M_service / S) / 1,000 MPa! Setting zero-moment top stress P_e / A - (P_e * e) / S = 0 reveals the Section Kern Point k = S / A: draped tendons placed below k (e > k) rely on self-weight gravity moment M_sw at transfer to keep the top flange in compression!
Verify on a Symmetric Prestressed Box/I-Girder—Gross_Area A = 0.24 m² (B2), Elastic_Modulus S = 0.040 m³ (B3 -> Kern Point k = 0.040 / 0.24 = 0.1667 m in E2), Effective_Prestress_Force P_e = 1,800.0 kN (B4), Midspan_Tendon_Eccentricity e = 0.25 m (B5), and Service_Gravity_Moment M = 500.0 kN·m (B6): Top Fiber Stress stays cleanly in compression at +8.75 MPa (E3), Bottom Fiber Precompression holds +6.25 MPa (E4, zero tension!), and Service Crack Class reads "ZERO-TENSION UNCRACKED" (E5)!
| Prestressed Girder KPI (A=0.24 m², S=0.04 m³, Pe=1,800 kN, e=0.25 m, M=500 kN·m) | Formula | Verified Output |
|---|---|---|
| Section Kern Radius k (m) | Section_Modulus_S_m3 / Area_A_m2 | 0.1667 m |
| Top-Fiber Service Stress (MPa) | (Pe/A - Pe*e/S + M/S) / 1000 | +8.75 MPa (+18.75 @ M = 900 kN·m) |
| Bottom-Fiber Service Stress (MPa) | (Pe/A + Pe*e/S - M/S) / 1000 | +6.25 MPa (-3.75 @ M = 900 kN·m!) |
| Service Tension Status | MIN(ftop, fbot) >= 0 -> ZERO-TENSION | ZERO-TENSION UNCRACKED |
Three-Term Elastic Superposition Bottom-Fiber Prestress Formula
Validate Area A > 0, Section_Modulus S > 0, Prestress P_e > 0, Eccentricity e >= 0, and Service_Moment M >= 0!
=LET(ar_m, B2, sm_m, B3, pe_k, B4, ec_m, B5, ms_k, B6, IF(OR(ar_m <= 0, sm_m <= 0, pe_k <= 0, ec_m < 0, ms_k < 0), "Invalid Prestress Input", ROUND(((pe_k / ar_m) + ((pe_k * ec_m) / sm_m) - (ms_k / sm_m)) / 1000, 2)))Heavy Live-Load Decompression & Bottom-Flange Tension Check (Set Service Moment M = 900 kN·m)
Increase Service_Moment B6 from 500.0 kN·m to 900.0 kN·m (heavy truck overload): Top Compression E3 climbs to +18.75 MPa while Bottom Precompression decompresses past zero into -3.75 MPa tension (E4), switching E5 to "TENSION ZONE CHECK CRACKING"!
=IF(ISTEXT(E4), E4, IF(AND(E3 >= 0, E4 >= 0), "ZERO-TENSION UNCRACKED", "TENSION ZONE CHECK CRACKING"))Try the example
Use the original values or change them to test your understanding.