EGB375 · Design of Concrete Structures
Prestressed Concrete — Mechanics, Allowable Stresses & Magnel Method
This chapter of EGB375 Design of Concrete Structures at Queensland University of Technology covers the elastic mechanics of prestressed concrete (PSC) beams: how an eccentric tendon produces top- and bottom-fibre stresses, how those stresses must satisfy code limits at both the transfer and service stages, and how the Magnel method turns four stress checks into four inequalities that bracket the initial prestress force. Everything follows the AS3600-2018 sign convention (tension positive, compression negative) and appears in the closed-book final exam.
What this chapter covers
- 01Model an eccentric tendon as an axial force −P/A plus a moment P·e about the section centroid
- 02Write the top- and bottom-fibre elastic stresses with the correct AS3600 sign convention (tension +ve)
- 03Distinguish the transfer stage (full prestress, self-weight only) from the service stage (reduced prestress, full load)
- 04Compute the transfer moment Mₒ and the short-term service moment Mₛ = W_G + 0.7W_Q
- 05State the four allowable stresses: f_ci and f_ti at transfer, f_c and f_t at service
- 06Assemble the Magnel four inequalities and read the feasible prestress region on the Magnel diagram
- 07Select an initial prestress Pᵢ inside the feasible band and verify every fibre stress
- 08Set out a parabolic (draped) tendon profile and find its drape, slope and vertical component
Magnel sizing of the initial prestress for a rectangular PSC beam
- +1Section properties: A = 350×800 = 280 000 mm²; I = bD³/12 = 1.4933×10¹⁰ mm⁴; symmetric so Z_t = Z_b = I/400 = 3.7333×10⁷ mm³.
- +1Self-weight W_sw = 25 kN/m³ × 0.280 m² = 7.0 kN/m, so the transfer moment Mₒ = W_sw L²/8 = 7.0×12²/8 = 126 kN·m.
- +1Service moment: permanent W_G = 7.0 + 6.0 = 13 kN/m; short-term combination Wₛ = W_G + 0.7W_Q = 13 + 0.7×10 = 20 kN/m, so Mₛ = 20×12²/8 = 360 kN·m.
- +1Allowable stresses: transfer f_ti = 0.25√32 = 1.414 MPa, f_ci = 0.5×32 = 16 MPa; service f_t = 0.6√40 = 3.795 MPa, f_c = 0.5×40 = 20 MPa.
- +1Coefficients (e = 330 mm): 1/A = 3.5714×10⁻⁶; e/Z = 8.8393×10⁻⁶; top k_t = e/Z − 1/A = 5.2679×10⁻⁶; bottom k_b = 1/A + e/Z = 12.4107×10⁻⁶ mm⁻². Mₒ/Z = 3.375 MPa, Mₛ/Z = 9.643 MPa.
- +1Magnel (1) transfer top ≤ f_ti: Pᵢ ≤ (1.414 + 3.375)/5.2679×10⁻⁶ = 909 kN (upper bound).
- +1Magnel (2) transfer bottom ≥ −f_ci: Pᵢ ≤ (16 + 3.375)/12.4107×10⁻⁶ = 1561 kN (upper, not governing).
- +1Magnel (3) service bottom ≤ f_t: Pᵢ ≥ (9.643 − 3.795)/(0.85×12.4107×10⁻⁶) = 554 kN (lower bound).
- +1Magnel (4) service top ≥ −f_c: requires Pᵢ ≥ (9.643 − 20)/(0.85×5.2679×10⁻⁶) < 0, so it is satisfied for any positive Pᵢ (not governing).
- +1Feasible band: 554 kN ≤ Pᵢ ≤ 909 kN; the transfer-top tension and service-bottom tension govern. Choose a mid-range Pᵢ = 700 kN.
- +1Verify transfer: σ_t = 700000×5.2679×10⁻⁶ − 3.375 = +0.31 MPa ≤ 1.414 ✓; σ_b = −700000×12.4107×10⁻⁶ + 3.375 = −5.31 MPa ≥ −16 ✓.
- +1Verify service: σ_b = −0.85×700000×12.4107×10⁻⁶ + 9.643 = +2.26 MPa ≤ 3.795 ✓; σ_t = 0.85×700000×5.2679×10⁻⁶ − 9.643 = −6.51 MPa ≥ −20 ✓. All four limits pass.
Key terms
- Eccentricity (e)
- Distance of the tendon below the section centroid (mm). It converts the prestress force P into an axial force plus a moment P·e, which is what pre-compresses the tension face.
- Section modulus (Z_t, Z_b)
- Z = I/y for each extreme fibre (mm³): Z_t = I/y_t to the top fibre, Z_b = I/y_b to the bottom. Equal only for a section symmetric about the centroid.
- Transfer stage
- The instant prestress is released into young concrete (strength f_cp). Only self-weight acts, so the top fibre tends to go into tension and the bottom into heavy compression.
- Service stage
- The in-use condition: concrete at full strength f′c, the full short-term load combination W_G + 0.7W_Q acting, and the prestress reduced to R·Pᵢ after time-dependent losses.
- Allowable stresses
- AS3600 limits: at transfer f_ci = 0.5f_cp (compression) and f_ti = 0.25√f_cp (tension); at service f_c = 0.5f′c and f_t = 0.25√f′c (or 0.6√f′c with bonded reinforcement).
- Magnel equations
- Four inequalities — two at transfer, two at service — obtained by forcing each fibre stress inside its allowable limit. Rearranged, each is a straight line of 1/Pᵢ against e.
- Magnel diagram
- A plot of 1/Pᵢ (vertical) versus e (horizontal). The four Magnel lines enclose a feasible region; any point inside it gives a prestress and eccentricity that satisfy every stress limit.
- Residual factor (R)
- The fraction of the initial prestress that survives time-dependent losses (creep, shrinkage, relaxation), typically 0.75–0.85. It scales the prestress used in the service-stage checks.
Prestressed Concrete — Mechanics, Allowable Stresses & Magnel Method FAQ
Why does a prestressed beam need checking at two different stages?
Because the worst tension face swaps. At transfer the prestress is full but the load is only self-weight, so the large P·e moment can crack the top fibre. At service the load is full but the prestress has decayed to R·Pᵢ, so the bottom fibre is the one at risk. A single check would miss one of them, which is exactly why the Magnel method uses two inequalities per stage.
What is the sign convention, and why does it matter so much?
Tension is positive and compression is negative. The axial prestress −P/A compresses both faces; the couple P·e tensions the top (+P·e/Z_t) and compresses the bottom; a sagging applied moment does the reverse. Reversing any one of these terms flips the whole check, which is the most common way marks are lost in this topic.
Can AI help me with prestressed concrete and the Magnel method in EGB375?
Yes — Sia can explain the mechanics step by step: how an eccentric tendon becomes an axial force plus a moment, how to set up the transfer and service fibre stresses, and how the four Magnel inequalities bracket the prestress. It works through the method and the reasoning so you can reproduce it under exam conditions; it does not sit the closed-book exam for you or promise a particular grade.
Studying with AI? Sia — free AI civil engineering tutor works through EGB375 step by step.
Exam move
Build fluency with the fibre-stress equations first: know cold that tension is positive, that the prestress couple tensions the top and compresses the bottom, and that a sagging moment does the opposite. Then drill the two-stage rhythm — transfer uses full Pᵢ with self-weight moment Mₒ and the young-concrete limits, service uses R·Pᵢ with the short-term moment Mₛ and the f′c limits. Practise assembling the four Magnel inequalities and reading the feasible band, because most exam questions are a variation on sizing Pᵢ for a given eccentricity. Since the final is closed-book with five A4 double-sided resource sheets you prepare yourself, write the fibre-stress equations, the four allowable-stress limits and a fully worked Magnel example onto your sheets in your own hand, and always finish by back-substituting your chosen prestress to verify every stress — that last check is where the final marks sit.
