Learn: Prestressed concrete fibre stress
Prestress produces a uniform axial stress and, when eccentric, a bending stress. An external bending moment adds another contribution, so the stress at one chosen fibre is found by adding three signed terms.
What the formula is saying
The terms are P/A, Pe y/I and M y/I. This lesson takes compression as positive; choose signed e, y and M so positive products add compression at the selected fibre. The signs come from a consistent section sketch, not from always adding magnitudes.
Read the symbols in plain language
- P
- Effective prestress
Effective prestress. Distribute the compressive prestress force over the section area.
NNewtons measure force; 1000 N = 1 kN.
- A
- Area
Area. Distribute the compressive prestress force over the section area.
m²Square metres measure area; square the length conversion factor.
- e
- Prestress eccentricity
Signed eccentricity from the section centroid to the prestress line, consistent with the chosen positive fibre coordinate.
mMetres measure length; 1 m = 1000 mm.
- y
- Fibre coordinate
Signed fibre distance from the centroid; positive and negative faces produce different bending stresses.
mMetres measure length; 1 m = 1000 mm.
- I
- Second moment
The area-weighted square of distance from the stated axis. It measures the spread of the cross-section, not its area or mass.
m⁴The fourth power of metres is used for a second moment of area; 1 m⁴ = 10¹² mm⁴.
- M
- Applied moment
Signed external moment; its contribution is added algebraically to P × e under this lesson’s convention.
N·mUse N·m as the base unit shown here. Use P in N, A in m², e and y in m, I in m⁴ and M in N·m. All three terms become Pa. y is the signed distance from the centroid to the evaluated fibre.
- σ
- Result to find
Prestressed concrete fibre stress. Superposition is valid for the stated uncracked linear-elastic section model.
Pa
Sort out the units first
Use P in N, A in m², e and y in m, I in m⁴ and M in N·m. All three terms become Pa. y is the signed distance from the centroid to the evaluated fibre.
Assumptions before calculating
Assume an uncracked linearly elastic section with plane sections remaining plane. P is the compressive prestress force at the stage considered, after whatever losses have already been included in that supplied value.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find σ and explain the result in the stated output unit.
- P · Effective prestress
- 1000000 N
- A · Area
- 0.3 m²
- e · Prestress eccentricity
- 0.15 m
- y · Fibre coordinate
- 0.3 m
- I · Second moment
- 0.02 m⁴
- M · Applied moment
- 200000 N·m
Find uniform prestress compression
Distribute the compressive prestress force over the section area.
(1000000) ÷ (0.3) ≈ 3333333.333 PaFind the eccentric-prestress contribution
The eccentric force creates a bending moment whose fibre stress depends on signed y.
(1000000) × (0.15) × (0.3) ÷ (0.02) = 2250000 PaFind the external-moment contribution
Use the external moment with a sign consistent with the chosen compression-positive convention.
(200000) × (0.3) ÷ (0.02) = 3000000 PaAdd the three signed stresses
Superposition is valid for the stated uncracked linear-elastic section model.
(3333333.333) + (2250000) + (3000000) ≈ 8583333.333 Pa
Does this worked answer make sense?
At the centroid y = 0, both bending terms vanish and stress is P/A. Switching from +y to −y reverses the bending terms but leaves uniform compression unchanged.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- P · Effective prestress
- 800000 N
- A · Area
- 0.25 m²
- e · Prestress eccentricity
- -0.12 m
- y · Fibre coordinate
- 0.25 m
- I · Second moment
- 0.015 m⁴
- M · Applied moment
- 100000 N·m
Find uniform prestress compression
Distribute the compressive prestress force over the section area.
(800000) ÷ (0.25) = 3200000 PaFind the eccentric-prestress contribution
The eccentric force creates a bending moment whose fibre stress depends on signed y.
(800000) × (-0.12) × (0.25) ÷ (0.015) = -1600000 PaFind the external-moment contribution
Use the external moment with a sign consistent with the chosen compression-positive convention.
(100000) × (0.25) ÷ (0.015) ≈ 1666666.667 PaAdd the three signed stresses
Superposition is valid for the stated uncracked linear-elastic section model.
(3200000) + (-1600000) + (1666666.667) ≈ 3266666.667 Pa
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- P · Effective prestress
- 1200000 N
- A · Area
- 0.3 m²
- e · Prestress eccentricity
- 0.1 m
- y · Fibre coordinate
- -0.3 m
- I · Second moment
- 0.02 m⁴
- M · Applied moment
- -150000 N·m
Find: Learn: Prestressed concrete fibre stress
A hint, not the answer
The terms are P/A, Pe y/I and M y/I. This lesson takes compression as positive; choose signed e, y and M so positive products add compression at the selected fibre. The signs come from a consistent section sketch, not from always adding magnitudes.
Use P in N, A in m², e and y in m, I in m⁴ and M in N·m. All three terms become Pa. y is the signed distance from the centroid to the evaluated fibre.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Find uniform prestress compression
Distribute the compressive prestress force over the section area.
(1200000) ÷ (0.3) = 4000000 PaFind the eccentric-prestress contribution
The eccentric force creates a bending moment whose fibre stress depends on signed y.
(1200000) × (0.1) × (-0.3) ÷ (0.02) = -1800000 PaFind the external-moment contribution
Use the external moment with a sign consistent with the chosen compression-positive convention.
(-150000) × (-0.3) ÷ (0.02) = 2250000 PaAdd the three signed stresses
Superposition is valid for the stated uncracked linear-elastic section model.
(4000000) + (-1800000) + (2250000) = 4450000 Pa
Avoid the common trap
Do not use unsigned eccentricity and fibre distance without a sign sketch. Do not mix gross and transformed section properties, or treat a positive result as a code acceptance check.
When this method applies — and when it does not
This does not compute prestress losses, secondary restraint moments, cracked-section redistribution, transfer strength or stress limits. A negative result indicates tension under the compression-positive convention and may invalidate the uncracked assumption.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Prestressed concrete fibre stress. First-generation EN 1992 teaching: material properties and the relevant bending, shear, serviceability, detailing or prestress relationship. Read the applicability conditions as well as the expression.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.