--- title: "Stepwise procedure of Mohr's circle" subject: "Strength of Materials" chapter: 4 key: "semester_3/strength_of_materials:mohr-circle-procedure" source: https://www.bctengineer.com/q/semester_3/strength_of_materials/mohr-circle-procedure site: BCT Engineering --- # Stepwise procedure of Mohr's circle Strength of Materials · Chapter 4: Stress and Strain Analysis - Asked in: 2081 Baisakh, 8 marks; 2068 Baisakh (old course), 8 marks (asked 2 times) - Video: [Mohr's Circle Steps](https://www.youtube.com/watch?v=tr0StsZsdpo) — The Open Educator - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/mohr-circle-procedure - Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-4 ## Question Write down the stepwise procedure (with sketch) for the determination of stresses in inclined plane for General stresses conditions ($\sigma_x$, $\sigma_y$ and $\tau_{xy}$ are given) using the Mohr's circle method. ## Short answer Draw σ-τ axes; plot X(σx, τxy) and Y(σy, −τxy) taking clockwise shear as positive. Centre C = ((σx+σy)/2, 0); radius R = √[((σx−σy)/2)² + τxy²]. Draw the circle. For an inclined plane, rotate radius CX by 2θ to read σn and τθ. Principal stresses: OC±R; τmax = R. ## Answer **Mohr's circle** is a graphical method that represents the state of plane stress at a point; every point on the circle gives the normal and shear stress on one plane. Given: $\sigma_x$, $\sigma_y$ and $\tau_{xy}$ on an element, and a plane whose normal is at angle $\theta$ to the $x$-axis. #### Procedure 1. Draw the $\sigma$-axis horizontally (tension positive, to the right of O) and the $\tau$-axis vertically through O. Select a suitable scale. 2. Sign rule for plotting shear: a shear stress that tends to turn the element **clockwise** is plotted **above** the $\sigma$-axis, and one that turns it anticlockwise is plotted below. With this rule, a rotation of $2\theta$ on the circle is in the same sense as the rotation $\theta$ of the plane. Plot point $X(\sigma_x, \tau_{xy})$ for the face on which $\sigma_x$ acts. 3. Plot point $Y(\sigma_y, -\tau_{xy})$ for the perpendicular face (its shear turns the element the opposite way). 4. Join X and Y. The line cuts the $\sigma$-axis at $C$, the centre, where $OC = \dfrac{\sigma_x+\sigma_y}{2}$. 5. With C as centre and $CX$ as radius, draw the circle. Radius $R = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}$. 6. The circle cuts the $\sigma$-axis at $P_1$ and $P_2$: $\sigma_1 = OC + R$, $\sigma_2 = OC - R$. The angle $XCP_1 = 2\theta_p$ gives the principal plane. 7. For the stresses on the given inclined plane, rotate the radius CX through $2\theta$ about C in the same sense as the plane is rotated, reaching point $Q$. The abscissa of Q is the normal stress $\sigma_n$ and the ordinate is the shear stress $\tau_\theta$. The diametrically opposite point $Q'$ gives the stresses on the perpendicular plane. 8. The resultant stress is $OQ$, and its obliquity is $\angle QOC$ measured from the $\sigma$-axis. 9. The top and bottom points of the circle give the maximum shear stress $\tau_{max} = R$. #### Sketch ```text tau | T (tau max = R) | .--*--. | / | \ | *Q | * X(sx, txy) --------+-P2-----C------P1------- sigma O \ | / | *Y | / | '--*--' | T' P2 = sigma_2, P1 = sigma_1, angle XCP1 = 2*theta_p ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which points are plotted first? A. X(sigma_x, 0) and Y(0, sigma_y) B. X(tau_xy, sigma_x) and Y(tau_xy, sigma_y) C. X(sigma_x, sigma_y) and Y(tau_xy, 0) D. X(sigma_x, tau_xy) and Y(sigma_y, -tau_xy) Answer: D. Each face is plotted as (normal stress, shear stress) with opposite shear signs. (Key point: Plotting the points) 2. What are the centre and radius? A. Centre (sigma_x - sigma_y)/2, radius tau_xy B. Centre OC = (sigma_x + sigma_y)/2, radius R = sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2) C. Centre at O, radius sigma_x D. Centre tau_xy, radius (sigma_x + sigma_y)/2 Answer: B. XY crosses the sigma-axis at the centre C, and CX is the radius. (Key point: Centre and radius) 3. How are stresses on a plane at theta found? A. Rotate CX through theta; ordinate is sigma_n B. Rotate CX through 2 theta; the radius is sigma_n C. Draw a line from O at angle theta D. Rotate CX through 2 theta about C in the same sense; the abscissa is sigma_n and the ordinate is tau Answer: D. The angle on the circle is double the physical angle. (Key point: Stresses on inclined plane) 4. Which statement is correct? A. sigma_1 = R, sigma_2 = -R, tau_max = OC B. sigma_1 = OC + R, sigma_2 = OC - R, and tau_max = R C. sigma_1 = OC, sigma_2 = R, tau_max = OC + R D. tau_max = 2R at the sigma-axis Answer: B. The circle meets the sigma-axis at the principal stresses; top and bottom points give tau_max = R. 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