CIV-21

Mohr 2-D principal stresses

σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]. The circle's far and near poles.

Reading speed

StressMohr 2-D

Governing equation

\sigma_{1,2}=\dfrac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}

where

\sigma_x: σx (MPa)
\sigma_y: σy (MPa)
\tau_{xy}: τxy (MPa)
\sigma_1: Algebraically larger (MPa)
\sigma_2: Algebraically smaller (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-21 — Mohr 2-D principal stresses) is the form associated with Mohr 2-D. Working symbols: $\sigma_x$ , $\sigma_y$ , $\tau_{xy}$ $\rightarrow$ $\sigma_1$ , $\sigma_2$ . Cauchy's 2-D eigenvalue problem. Plane stress or plane strain in the x–y face.

Purpose

Purpose: compute

\sigma_1

\sigma_2

from

\sigma_x

\sigma_y

\tau_{xy}

in Structural & civil via

\sigma_{1,2}=\dfrac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}

σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]. The circle's far and near poles. 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

\sigma_x = 80.000\,\mathrm{MPa}

\sigma_y = 20.000\,\mathrm{MPa}

\tau_{xy} = 30.000\,\mathrm{MPa}

, the governing relation

\sigma_{1,2}=\dfrac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}

yields

\sigma_1 = 92.43\,\mathrm{MPa}

\sigma_2 = 7.57\,\mathrm{MPa}

. A stress element, a Mohr circle, two principal ticks. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

σx \sigma_x80 MPaσy \sigma_y20 MPaτxy \tau_{xy}30 MPa

Outputs

Algebraically larger \sigma_192.43 MPa
Algebraically smaller \sigma_27.57 MPa

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A stress element, a Mohr circle, two principal ticks.

Cauchy's 2-D eigenvalue problem. Plane stress or plane strain in the x–y face.

Reading speed

Watch on YouTube