Prestress Friction Loss
Calculate the prestress force loss from friction along a curved tendon, ΔP = P_0·(1 − e^(−(μα + k·x))), from the jacking force P_0 (kN), the tendon-duct friction coefficient μ, the sum of tendon deviation angles α (radians), the wobble coefficient k (loss per metre, 1/m) and the tendon length x (m). In POST-TENSIONING (where the tendon is tensioned after the concrete hardens, sliding inside a duct embedded in the member), the force applied at the end by the jack does NOT arrive full at the other end: FRICTION between tendon and duct consumes part of it along the path. There are two effects: friction in the tendon CURVES (μα term — the more the tendon curves, the more it 'squeezes' the duct and the greater the friction, like a rope on a pulley — the capstan effect) and 'wobble' friction in straight runs (k·x term — from small undulations and duct misalignment). Friction loss makes the prestress force DECREASE progressively from the active end (jack) to the passive (dead anchorage), which is why long tendons are sometimes tensioned from BOTH ends. It is an immediate loss, computed tendon by tendon. Enter the jacking force, friction coefficient, sum of angles, wobble coefficient and length.
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Prestress friction loss
The friction loss of prestressing force along a curved tendon is ΔP = P_0·(1 − e^(−(μα + k·x))), from the force applied at the jack P_0, the tendon-duct friction coefficient μ, the sum of the angular deviations of the tendon α (radians), the wobble friction coefficient k (loss per metre) and the tendon length x. In post-tensioned prestressing (where the tendon is stressed after the concrete has hardened, sliding inside a duct cast into the member), the force applied at one end by the jack never arrives intact at the far end: friction between tendon and duct consumes part of it along the way. Two effects act together: friction at the curves of the tendon (term μα — the sharper the curvature, the harder the tendon presses against the duct and the greater the friction, like a rope on a pulley — the capstan effect) and wobble friction along the straight runs (term k·x — caused by small undulations and placement imperfections of the duct). Friction loss makes the prestressing force fall progressively from the live end (jack) toward the passive end (dead anchorage), and that is why long tendons are sometimes stressed from both ends. It is an immediate loss, computed tendon by tendon. Enter the applied force, the friction coefficient, the sum of the angles, the wobble coefficient and the length.
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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.
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.
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.
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.
Friction Torque
Calculate the friction torque in a shaft or bearing, T = μ·F·r, multiplying the friction coefficient μ by the normal force (load) F and the radius r where friction acts. The result, in N·m, is the moment friction opposes to rotation — the torque the motor must overcome just to turn the assembly, without doing useful work. Reducing friction torque (with lubrication, rolling bearings and good finishes) saves energy and lowers heating. Multiplied by the angular velocity, it gives the power dissipated by friction. Enter the friction coefficient, the force and the radius.
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