Elastic Shortening Loss
Calculate the prestress loss from concrete elastic shortening, Δσ = (E_s/E_c)·σ_c, from the steel modulus E_s (MPa), the concrete modulus E_c (MPa) and the concrete stress at the tendon level σ_c (MPa). It is one of the IMMEDIATE prestress losses (at transfer, not over time): when the tendon is tensioned and anchored, it compresses the concrete, and the concrete, being compressed, SHORTENS elastically. Since the tendon is bonded or anchored in this shortened concrete, it shortens too — and shortening, it LOSES part of its tension. The loss is proportional to the modular ratio αe = E_s/E_c (typically 6-8, since steel is much stiffer than concrete) times the concrete compression stress at the tendon level. In members with SEVERAL tendons prestressed sequentially, each new tendon compresses and shortens the concrete, causing loss in already-anchored tendons — so the average loss is often taken as half the value (the first tendons lose more than the last). This is one of the losses to subtract from the initial force to get the effective prestressing force. Enter the steel and concrete moduli and the concrete stress.
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Elastic shortening loss
The prestress loss from elastic shortening of the concrete is Δσ = (E_s/E_c)·σ_c, from the modulus of elasticity of the steel E_s, the modulus of the concrete E_c and the concrete stress at the tendon level σ_c. It is one of the immediate losses (it happens at the moment of stressing, never over time): when the tendon is stressed and anchored, it compresses the concrete, and the concrete, once compressed, shortens elastically. Since the tendon is bonded to or anchored in that shortened concrete, it shortens along with it — and by shortening it gives up part of its tension (it slackens slightly). The loss is proportional to the modular ratio αe = E_s/E_c (typically 6 to 8, since steel is far stiffer than concrete) times the compressive stress in the concrete at the height of the tendon. In members with several tendons stressed in sequence, every new tendon compresses and shortens the concrete, causing a loss in the tendons already anchored — which is why the average loss is often taken as half the computed value (the first tendons lose more than the last ones). This is one of the losses the design must deduct from the initial force to arrive at the effective prestressing force. Enter the moduli of elasticity of the steel and of the concrete and the stress in the concrete.
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Concrete Creep Loss
Calculate the prestress loss from concrete creep, Δσ = φ·(E_s/E_c)·σ_cg, from the creep coefficient φ (dimensionless), the modular ratio E_s/E_c and the concrete stress at the tendon level from permanent loads σ_cg (MPa). Creep is the SLOW, growing deformation concrete undergoes under CONSTANT load over time: besides the immediate elastic shortening when compressed, concrete keeps shortening gradually for months and years, reaching a total deformation 2-3 times the initial elastic one. In a prestressed member, the concrete is PERMANENTLY compressed by the prestress, so it creeps (shortens slowly), and the bonded tendon shortens with it, LOSING tension — the largest time-dependent loss in many cases. The loss is the creep coefficient φ (typically 1.5-3.5, a function of humidity, loading age, member dimensions) times the equivalent elastic loss (modular ratio × concrete stress). With shrinkage and relaxation, creep defines the total time-dependent prestress loss. Estimating these losses well is crucial: underestimating leaves the member with less prestress than intended (cracking risk); overestimating wastes steel. Enter the creep coefficient, the modular ratio and the concrete stress.
Steel Relaxation Loss
Calculate the prestress loss from steel relaxation, Δσ = (ψ/100)·σ_pi, from the relaxation coefficient ψ (% of initial stress) and the initial tendon stress σ_pi (MPa). Relaxation is a STEEL phenomenon analogous to concrete creep: when a steel wire or strand is held under CONSTANT tension (fixed elongation, as in an anchored prestressing tendon), its stress DECREASES slowly over time, even without length change. It is as if the steel 'yields' microscopically under prolonged load, losing part of its tension. Relaxation depends on the steel type (LOW-relaxation steels — LR —, thermomechanically treated, relax much less, ~2.5% in 1000h at 0.7·fptk, than normal-relaxation — NR —, ~12%), the initial stress level (the higher, the more relaxation) and temperature. The coefficient ψ is tabulated as a function of these factors and time. Relaxation is one of the TIME-DEPENDENT prestress losses, along with concrete shrinkage and creep, and design sums them all for the total loss and the effective final prestressing force. Enter the relaxation coefficient and the initial tendon stress.
Concrete Shrinkage Loss
Calculate the prestress loss from concrete shrinkage, Δσ = ε_cs·E_s, from the shrinkage strain ε_cs (dimensionless) and the steel modulus E_s (MPa). Shrinkage is the volume reduction concrete undergoes over time as it LOSES water by evaporation (drying shrinkage) and through cement hydration reactions (autogenous shrinkage), independent of loading. When the concrete of a prestressed member shrinks (shortens), the bonded steel tendon shortens too — and shortening, it LOSES tension, exactly as in elastic-shortening loss, except here the shortening is from shrinkage and occurs SLOWLY over months and years. The loss is simply the shrinkage strain times the steel modulus (the stress that shortening 'steals' from the tendon). The shrinkage strain ε_cs is typically 0.0002-0.0005 (200-500 microstrains) and depends on ambient humidity (drier = more shrinkage), member dimensions (thin members shrink more, losing water faster), mix and time. It is one of the three time-dependent losses (with creep and relaxation) reducing prestress over the structure's life. Enter the shrinkage strain and the steel modulus.
Final Prestressing Force
Calculate the final (effective) prestressing force after losses, P_∞ = P_0·(1 − losses/100), from the initial prestressing force P_0 (kN) and the total loss percentage (%). A member's prestressing force is NOT constant: it starts at an initial value (the jacking force) and DECREASES due to the various losses — immediate (elastic shortening, friction, anchorage set) and time-dependent (concrete shrinkage and creep, steel relaxation). The effective final force, after all losses stabilize (after years), is what actually acts in the structure in service and must ensure performance. Total losses typically sum 15-25% of the initial force in post-tensioned structures and can reach 20-30% in pretensioned ones. This simple calculation applies the total loss percentage to the initial force, giving the effective force — fundamental to check service stresses, cracking and member deflection. The designer works with TWO critical situations: maximum INITIAL force (right after transfer, member still unloaded — risk of excess compression or top-fiber tension) and minimum FINAL force (after all losses, member loaded — risk of decompression and cracking). Enter the initial force and the total loss percentage.
Edge Stress from Prestressing
Calculate the normal stress at an extreme fiber of a prestressed concrete section, σ = P/A + (P·e)/W, from the prestressing force P (MN), the section area A (m²), the tendon eccentricity e (m) and the section modulus W (m³). Prestressed concrete is one of the great structural engineering inventions of the 20th century: high-strength steel tendons are tensioned (prestressed) and anchored in the member, COMPRESSING the concrete before it even receives service loads. Since concrete is strong in compression but weak in tension, this pre-compression 'cancels' the tensions external loads would cause, allowing much longer spans and slenderer members than conventional reinforced concrete. The tendon is placed with ECCENTRICITY (below the centroid), so prestressing generates not only uniform compression (P/A) but also a moment (P·e) producing stresses opposite to the loading — compressing exactly the fiber that would tend to crack. This formula computes the resulting edge stress, summing axial compression and prestress bending; design verifies stresses stay within limits in all phases (at transfer, empty, and in service, loaded). Enter the prestressing force, area, eccentricity and section modulus.
Prestress Moment
Calculate the moment generated by eccentric prestressing at a section, M_p = P·e, from the prestressing force P (kN) and the tendon eccentricity e (m). When the prestressing tendon is positioned with ECCENTRICITY relative to the section centroid (usually below, in the region tensioned by loads), the prestressing force, besides axially compressing the section (P/A), generates a BENDING MOMENT equal to force times eccentricity. This prestress moment is the key to prestressed concrete's efficiency: it is OPPOSITE to the moment from external loads (self-weight, live loads), 'bowing' the member upward (camber) and producing top-fiber tension and bottom-fiber compression — exactly the opposite of what the load does. So eccentric prestressing 'pre-loads' the member against the service loading, so that when loads act, they must first CANCEL the prestress effects before tensioning the concrete. That is why prestressed beams often show camber (upward curvature) when still unloaded. The prestress moment is fundamental in computing edge stresses, camber and the optimal tendon profile along the member (which roughly follows the load moment diagram, with varying eccentricity). Enter the prestressing force and the eccentricity.
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