CIV-38

Biaxial Mohr: τmax and θp

τmax = √[((σx−σy)/2)² + τxy²], θp = ½ atan2(2τxy, σx−σy).

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StressMohr biaxial

Governing equation

\tau_{\max}=\sqrt{\bigl((\sigma_x-\sigma_y)/2\bigr)^2+\tau_{xy}^2},\quad \theta_p=\tfrac12\mathrm{atan2}(2\tau_{xy},\sigma_x-\sigma_y)

where

\sigma_x: σx (MPa)
\sigma_y: σy (MPa)
\tau_{xy}: τxy (MPa)
\tau_{max}: Max in-plane shear (MPa)
\theta_p: Principal angle (°)

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-38 — Biaxial Mohr: τmax and θp) is the form associated with Mohr biaxial. Working symbols: $\sigma_x$ , $\sigma_y$ , $\tau_{xy}$ $\rightarrow$ $\tau_{max}$ , $\theta_p$ . The radius of Mohr's circle is the maximum in-plane shear. Principal axes sit 2θp on the circle.

Purpose

Purpose: compute

\tau_{max}

\theta_p

from

\sigma_x

\sigma_y

\tau_{xy}

in Structural & civil via

\tau_{\max}=\sqrt{\bigl((\sigma_x-\sigma_y)/2\bigr)^2+\tau_{xy}^2},\quad \theta_p=\tfrac12\mathrm{atan2}(2\tau_{xy},\sigma_x-\sigma_y)

τmax = √[((σx−σy)/2)² + τxy²], θp = ½ atan2(2τxy, σx−σy). 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 = 90.000\,\mathrm{MPa}

\sigma_y = -10.000\,\mathrm{MPa}

\tau_{xy} = 40.000\,\mathrm{MPa}

, the governing relation

\tau_{\max}=\sqrt{\bigl((\sigma_x-\sigma_y)/2\bigr)^2+\tau_{xy}^2},\quad \theta_p=\tfrac12\mathrm{atan2}(2\tau_{xy},\sigma_x-\sigma_y)

yields

\tau_{max} = 64.03\,\mathrm{MPa}

\theta_p = 19.3\,\mathrm{^{\circ}}

. A circle, a radius τmax, an angle 2θp. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

σx \sigma_x90 MPaσy \sigma_y-10 MPaτxy \tau_{xy}40 MPa

Outputs

Max in-plane shear \tau_{max}64.03 MPa
Principal angle \theta_p19.3 °

Reading speed

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Narration of this film

A circle, a radius τmax, an angle 2θp.

The radius of Mohr's circle is the maximum in-plane shear. Principal axes sit 2θp on the circle.

Reading speed

Watch on YouTube