Prestressed Beam Losses Calculator

Understanding prestress losses

Introduction: Why prestress losses matter

Prestressed concrete works because the tendon force introduces compression into the concrete member, offsetting tensile stresses from self-weight and service loads. In practice, the force you apply at jacking (or the force at transfer for pretensioned members) is not the force that remains. A combination of immediate and time-dependent mechanisms reduces tendon force and tendon stress. If losses are underestimated, the member may crack earlier than expected, deflections may increase, and long-term durability can suffer. If losses are overestimated, the design may become uneconomical due to excessive strand quantity or higher jacking forces.

This calculator provides a simplified estimate of four common loss components: elastic shortening, creep, shrinkage, and steel relaxation. These are often the dominant contributors in many preliminary studies. The output is useful for building intuition: you can see which input parameter drives the effective prestress most strongly and how sensitive the result is to creep and shrinkage assumptions.

What this calculator includes (and what it does not)

The model intentionally stays compact so it can run in a single file and remain easy to audit. It includes: elastic shortening loss (a proportional estimate), creep loss (as a multiple of elastic loss), shrinkage loss (from shrinkage strain compatibility), and relaxation loss (as a percentage of initial force).

It does not explicitly compute friction losses in post-tensioning ducts, anchorage seating (wedge draw-in), curvature/wobble effects, temperature effects, staged stressing, composite action, or changes in section properties. Those effects can be significant—especially for post-tensioned members with long tendons and curved profiles. A common workflow is to reduce the initial force for friction and seating first, then apply time-dependent losses to the reduced force.

How to use this calculator

1. Enter the initial prestress force P_i in kN. Depending on your convention, this may be the jacking force or the force immediately after transfer.
2. Enter areas in mm²: prestressing steel area A_ps and concrete area A_c. For the simplified elastic-shortening estimate, A_c is typically the gross concrete area of the member.
3. Enter elastic moduli in GPa: steel E_s and concrete E_c. The calculator converts GPa to MPa internally.
4. Provide a creep coefficient φ, shrinkage strain ε_sh in microstrain (με), and relaxation loss as a percent of P_i.
5. Press Calculate to see each loss component, the effective force P_eff, and effective steel stress f_pe.

Units are critical. This page assumes: P_i in kN, areas in mm², moduli in GPa, shrinkage in με, and relaxation in percent. If you use other units, convert them before entering values. The effective stress output is in MPa because the internal force is in N and area is in mm² (1 N/mm² = 1 MPa).

Formulas used (simplified model)

The following relationships are used exactly as implemented in the script. They are simplified and assume a prismatic member with uniform stress distribution. Symbols match the input labels.

Elastic shortening loss

$$\mathrm{\Delta P}{\mathrm{es}}_{}=P{i}_{}·\frac{A}{{\mathrm{ps}}_{}}\frac{E}{c_{}}$$ where P_i is initial prestressing force, A_ps is prestressing steel area, A_c is concrete area, and E_c, E_s are elastic moduli.

Creep loss

$$\mathrm{\Delta P}{\mathrm{cr}}_{}=\phi \mathrm{\Delta P}{\mathrm{es}}_{}$$ The creep coefficient φ depends on age at loading, humidity, member size, and mix design. In many practical situations, creep is one of the largest uncertainties in long-term prestress prediction.

Shrinkage loss

$$\mathrm{\Delta P}{\mathrm{sh}}_{}=\epsilon {\mathrm{sh}}_{}E{s}_{}A{\mathrm{ps}}_{}$$ with ε_sh entered in microstrain (με) and converted to decimal strain. Shrinkage is strongly influenced by curing quality, cement content, member thickness, and ambient humidity.

Relaxation loss

$$\mathrm{\Delta P}{\mathrm{rel}}_{}=rP{i}_{}$$ where r is the relaxation fraction (e.g., 2% → 0.02). Relaxation depends on tendon type (low-relaxation vs. stress-relieved), temperature, and sustained stress level.

Total effective prestress

$$P{\mathrm{eff}}_{}=P{i}_{}-\mathrm{\Delta P}{\mathrm{es}}_{}+\mathrm{\Delta P}{\mathrm{cr}}_{}+\mathrm{\Delta P}{\mathrm{sh}}_{}+\mathrm{\Delta P}{\mathrm{rel}}_{}$$ The effective steel stress is f_pe = P_eff / A_ps.

Worked example (step-by-step)

Use the default inputs to reproduce a quick example. Assume a pretensioned girder with: P_i = 1000 kN, A_ps = 1500 mm², A_c = 300,000 mm², E_s = 200 GPa, E_c = 30 GPa, creep coefficient φ = 1.6, shrinkage strain ε_sh = 600 με, and relaxation = 2%.

First compute elastic shortening loss: the area ratio is 1500/300,000 = 0.005, and the modulus ratio is 30/200 = 0.15. The product is 0.00075, so ΔP_es ≈ 1000 kN × 0.00075 = 0.75 kN in this simplified formulation. Next, creep loss is ΔP_cr = 1.6 × 0.75 kN ≈ 1.2 kN. Shrinkage loss uses strain compatibility: 600 με = 0.000600, so ΔP_sh = 0.000600 × 200,000 MPa × 1500 mm² ≈ 180,000 N = 180 kN. Relaxation loss is 2% of 1000 kN, so ΔP_rel = 20 kN.

Total loss is approximately 0.75 + 1.2 + 180 + 20 = 201.95 kN, giving P_eff ≈ 798.1 kN. The effective stress is f_pe ≈ 798.1 kN / 1500 mm² = 532 MPa. Your results may differ slightly due to rounding. The key takeaway is that, for these inputs, shrinkage dominates the loss estimate.

Assumptions, interpretation, and limitations

• Simplified mechanics: The elastic-shortening relationship used here is a proportional estimate. It does not include tendon eccentricity, section modulus effects, composite sections, staged construction, or varying stress along the member.
• Not a code check: This page is not a substitute for AASHTO LRFD, Eurocode 2, ACI, or other governing code procedures. Use code-based methods and project-specific parameters for final design.
• Omitted losses: Friction losses in post-tensioning ducts, anchorage seating/slip, curvature/wobble effects, temperature gradients, and construction tolerances are not included.
• Input sensitivity: Shrinkage and creep can dominate results. If you are comparing alternatives, keep assumptions consistent (same humidity class, curing regime, and age at loading).
• Meaning of “effective”: The output P_eff is the initial force minus the sum of the modeled losses. In real projects, “effective” may be defined at a specific time (e.g., at service, at 1000 hours, at 1 year) and may include additional loss terms.
• Sanity checks: If the effective force becomes negative, it indicates inconsistent inputs (for example, extremely high shrinkage strain or relaxation percentage). In practice, designers would revisit assumptions, tendon layout, and construction sequence.

Typical parameter ranges (context)

The ranges below are common for preliminary design and for building intuition. Replace them with project-specific values whenever possible. For example, a dry climate with thin webs and minimal curing can push shrinkage toward the upper end, while a humid environment with good curing and larger member thickness can reduce shrinkage.

Practical guidance for better inputs

If you are using this calculator for a real project, the quality of the result depends on the quality of the inputs. Consider the following practical steps. For creep and shrinkage, use values consistent with your governing code and your assumed environment (humidity, curing duration, member size). For relaxation, use manufacturer data for the specific strand or bar type and the expected stress level. For the concrete modulus, use a value consistent with the concrete strength at the time of transfer or loading. If you are unsure, run a sensitivity study: try a low, mid, and high value for φ and ε_sh and observe how much P_eff changes.

Finally, remember that prestress losses are only one part of serviceability. Effective prestress interacts with section properties, tendon eccentricity, dead load moments, live load distribution, and cracking criteria. Use the effective stress output as an input to your broader service stress checks and deflection estimates.