CIV-36

Prestress combined stress

σ = P/A ± P e c / I. Decompression when the tensile bit cancels P/A.

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PrestressMagnel

Governing equation

\sigma=P/A\pm Pec/I

where

P: Prestress force (kN)
A: Area (cm^2)
e: Eccentricity (mm)
c: Fibre distance (mm)
I: Second moment (cm^4)
\sigma_{top}: Top fibre (MPa)
\sigma_{bot}: Bottom fibre (MPa)

Lecture brief

Historical brief

From Euler’s 1744 elastica and Navier’s beam theory to Mohr’s circle and transformed-section RC, structural mechanics grew as a closed-form craft before finite elements. The lab keeps those governing lines for buckling, flexure, joints and influence. This sheet (CIV-36 — Prestress combined stress) is the form associated with Magnel. Working symbols: $P$ , $A$ , $e$ , $c$ , $I$ $\rightarrow$ $\sigma_{top}$ , $\sigma_{bot}$ . A concentric P is uniform compression. An eccentricity e is a moment Pe, superimposed elastically.

Purpose

Purpose: compute

\sigma_{top}

\sigma_{bot}

from

P

A

e

c

I

in Structural & civil via

\sigma=P/A\pm Pec/I

σ = P/A ± P e c / I. Decompression when the tensile bit cancels P/A. Use it when a real structural & civil question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

P = 1500.000\,\mathrm{kN}

A = 2000.000\,\mathrm{cm^2}

e = 120.000\,\mathrm{mm}

c = 300.000\,\mathrm{mm}

I = 180000.000\,\mathrm{cm^4}

, the governing relation

\sigma=P/A\pm Pec/I

yields

\sigma_{top} = -22.50\,\mathrm{MPa}

\sigma_{bot} = 37.50\,\mathrm{MPa}

. A concrete section, a tendon offset, two fibre stresses. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Prestress force P1500 kNArea A2000 cm^2Eccentricity e120 mmFibre distance c300 mmSecond moment I180000 cm^4

Outputs

Top fibre \sigma_{top}-22.50 MPa
Bottom fibre \sigma_{bot}37.50 MPa

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Narration of this film

A concrete section, a tendon offset, two fibre stresses.

A concentric P is uniform compression. An eccentricity e is a moment Pe, superimposed elastically.

Reading speed

Watch on YouTube