CIV-03

Mohr principal stresses

In-plane principals from σx, σy and τxy.

Reading speed

Stress stateMohr 1882EN 1993-1-1

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: Normal σx (MPa)
\sigma_y: Normal σy (MPa)
\tau_{xy}: Shear τxy (MPa)
\sigma_1: Major principal (MPa)
\sigma_2: Minor principal (MPa)
\tau_{\max}: Max shear (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-03 — Mohr principal stresses) is the form associated with Mohr 1882 · EN 1993-1-1. Working symbols: $\sigma_x$ , $\sigma_y$ , $\tau_{xy}$ $\rightarrow$ $\sigma_1$ , $\sigma_2$ , $\tau_{\max}$ . Mohr's circle has centre (σx+σy)/2 and radius √(((σx−σy)/2)²+τ²). The principals are the horizontal intercepts.

Purpose

Purpose: compute

\sigma_1

\sigma_2

\tau_{\max}

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}

In-plane principals from σx, σy and τxy. 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 = 92.426\,\mathrm{MPa}

\sigma_2 = 7.574\,\mathrm{MPa}

\tau_{\max} = 42.426\,\mathrm{MPa}

. 2-D plane stress. Sign convention: tension positive. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Normal σx \sigma_x80 MPaNormal σy \sigma_y20 MPaShear τxy \tau_{xy}30 MPa

Outputs

Major principal \sigma_192.426 MPa
Minor principal \sigma_27.574 MPa
Max shear \tau_{\max}42.426 MPa

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Narration of this film

2-D plane stress. Sign convention: tension positive.

Mohr's circle has centre (σx+σy)/2 and radius √(((σx−σy)/2)²+τ²). The principals are the horizontal intercepts.

Reading speed

Watch on YouTube