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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Perda por atrito na protensão
A perda de força de protensão por atrito ao longo de um cabo curvo é ΔP = P_0·(1 − e^(−(μα + k·x))), a partir da força aplicada no macaco P_0, do coeficiente de atrito cabo-bainha μ, da soma dos ângulos de desvio do cabo α (radianos), do coeficiente de atrito parasita k (perda por metro) e do comprimento do cabo x. Na protensão pós-tracionada (em que o cabo é tracionado depois de o concreto endurecer, deslizando dentro de uma bainha embutida na peça), a força aplicada na extremidade pelo macaco não chega integral à outra ponta: o atrito entre o cabo e a bainha consome parte dela ao longo do percurso. Há dois efeitos: o atrito nas curvas do cabo (termo μα — quanto mais o cabo curva, mais ele 'aperta' a bainha e maior o atrito, como uma corda numa polia — o efeito capstan) e o atrito 'parasita' nos trechos retos (termo k·x — devido a pequenas ondulações e imperfeições de posicionamento da bainha). A perda por atrito faz a força de protensão diminuir progressivamente da extremidade ativa (macaco) para a passiva (ancoragem morta), e é por isso que cabos longos às vezes são protendidos pelas duas extremidades. É uma perda imediata, calculada cabo a cabo. Informe a força aplicada, o coeficiente de atrito, a soma dos ângulos, o coeficiente parasita e o comprimento.
Related Tools
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.
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